A President has absolute immunity from civil liability for his official acts. As discussed here (citations omitted):
Although the President is sued daily in his governmental capacity, he
normally is not sued in his personal capacity, which would hold him
personally liable. In 1982, the Supreme Court held in Nixon v.
Fitzgerald that the President enjoys absolute immunity from civil
litigation for official acts undertaken while he or she is President.
The Court suggested that this immunity was broad (though not
limitless), applying to acts within the "outer perimeter'" of the
President's official duties. Fifteen years after Fitzgerald, the
Supreme Court held in Clinton v. Jones that the President does not
possess absolute immunity from civil litigation surrounding acts he
took before becoming President. The 2020 Supreme Court decision in
Trump v. Vance held that the president is subject to subpoenas in criminal prosecutions for personal conduct with the same legal
threshold as anyone else.
So, one of the first legal issues that a court would have to confront is whether the acts in question were within the scope of the President's official acts and duties. If so, there is immunity. If this act is determined by the Court to be a private act of the President or ultra vires (beyond a reasonable view of the scope of his official duties), then absolute immunity would not apply.
This would be analogous to the case of Carroll v. Trump (discussed, for example here) in which Carroll was allegedly defamed by statements made by Trump in an unofficial capacity and his civil action for defamation against Trump was, at least at some stages, allowed to proceed because it was outside the scope of Presidential absolute immunity.
On October 26, 2020, the trial court in the Carroll v. Trump case held in the linked opinion, that Presidential absolute immunity did not apply to that claim.
The analysis would be much less clear, although it might come up the same way, in a wrongful death case brought by Sicknick's next of kin as authorized by statute (presumably on a common law claim for assault resulting in a wrongful death, or for civil conspiracy to commit an assault resulting a wrongful death). This is because, unlike Carroll which pertained to statements made by Trump about Trump's pre-Presidential conduct, Sicknick would pertain the conduct by Trump while President, in relation to a federal election in which he was a Presidential candidate while holding Presidential office, and the actions that he urged the Capitol rioters to take to respond to the imminent Congressional certification of the results of that election which he claimed (without any factual basis and after having had the claim rejected by approximately five dozen courts) was fraudulently obtained.
Qualified immunity is a doctrine to liability arising under 42 U.S.C. § 1983, which authorizes civil actions for violations of constitutional rights. There is § 1983 liability only when someone intentionally deprives another of their constitutional rights under color of law, and causes injury thereby. Qualified immunity states the principle that even if there is a deprivation of constitutional rights, a government official has liability only if the right is clearly established under applicable law.
The key issue in this claim would be whether the President's actions were made "under color of law" and/or whether they resulted in a deprivation of rights "under color of law".
The answer in this case is probably "no." The people who caused Officer Brian Sicknick to suffer injuries in January 6, 2021 incident at the Capitol building that caused his death were not purporting to acts in furtherance of authority granted to them by the U.S. government or any other legitimate government, so this was not a civil rights violation, even though it may have been tortious conduct.
Any criminal sanction would have to arise either under federal law (sedition would be the obvious charge) or perhaps under D.C.'s penal code (solicitation of murder, or conspiracy to commit murder would be possible charges).
In general, immunity from criminal charges would probably be subjected to a similar absolute immunity analysis, although there is very little case law on point because there are almost no examples of Presidents or former Presidents facing federal criminal charges (although Nixon v. Fitzgerald and Trump v. Vance come close, although Trump v. Vance is complicated by that fact that it involves state law criminal charges).
In particular, the U.S. Supreme Court in Nixon v. Fitzgerald, the Court emphasized in dicta in its 5-4 decision that the President is not immune from criminal charges stemming from his official or unofficial acts while he is in office. But certainly acts that would be criminal if authorized by someone else, like ordering the CIA to attempt to kill a suspected terrorist, which are without the scope of the President's official acts, would not give rise to criminal liability, so this dicta is something of an overstatement.
One could argue, however, that in a case like that one (and most acts that would otherwise constitute crimes that are legal when authorized by the President), there is not governmental immunity from criminal liability, and that there is instead a "justification" for the otherwise criminal act that is a non-immunity based exception to criminal liability similar to the basis upon which a police officer has an affirmative justified use of force defense to a murder charge when he used force authorized by law to make prevent a crime or stop a violent felony who is a danger to the public from escaping.
If the immunity analysis is correct, then there is criminal liability if this act is determined by the Court to be a private act of the President or ultra vires (beyond a reasonable view of the scope of his official duties), a criminal prosecution could proceed, and this question would be resolved by the judge in a pre-trial hearing.
If the "justification" rather than the immunity analysis is correct, then the question would be whether there was any legalized justification under the law for the acts alleged to amount to criminal conduct that the President was charged with in the indictment, either on the face of the indictment, if the facts alleged there are clear enough (in which case it could be resolved by a judge in a pre-trial hearing) or as applied to the specific facts of the case (in which case the question of justification would be resolved by the jury at trial). It is not obvious what common law or statutory theory could justify the acts alleged.
If there was no valid affirmative defense of legal justification to the charges, and the Courts held that there was no such thing as absolute immunity from criminal prosecutions for official acts, then the core of the criminal case would focus on proving the statutory elements of the charges brought and overcoming the First Amendment affirmative defense discussed below.
An analysis of some of the pertinent murder charges can be found in this Law.StackExchange answer, albeit without the additional layer of immunity analysis attached. An analysis of the pertinent sedition statutes can be found in this Law.StackExchange answer, again, without the additional lawyers of immunity analysis.
As a practical matter and as a matter of Justice Department policy, a President may not be prosecuted for federal crimes while in office. But that isn't a bar to criminal prosecution of a President for actions taken by the President outside of his official duties while in office (or alternately, because there is no lawful justification for his actions).
For example, if the President assaulted his wife in a domestic violence incident that took place in the White House, he could be prosecuted for that crime by the D.C. prosecutor at any time after his term of office expired but before the statute of limitations had run.
The President could in theory also be prosecuted for this conduct if absolute immunity did not apply (or does not exist as a matter of law for criminal conduct, a question is has not authoritatively been resolved although persuasive authority from other cases suggests that it may not). No one could compel either federal or D.C. prosecutors to do so, however. Any such prosecution would be in the sole discretion of the pertinent prosecutors.
Further, since these would be felony prosecutions under federal law (D.C. law is also a special subset of federal law), the prosecutions could only be brought pursuant to a grand jury convened either by a Justice Department attorney in the U.S. District Court for the District of Columbia, or by a District of Columbia government appointed prosecutor in the Criminal Division of the Superior Court of the District of Columbia.
In either case, however, both the grand jury whose approval would be required to indict Trump, and the petite jury trying the case on the merits, would be drawn from the District of Columbia, which is pretty much the least favorable jury pool that Donald Trump could imagine. There is no state in the United States where a larger share of the Presidential election was cast against Donald Trump than in the District of Columbia.
Of course, criminal liability primarily imposed a sentence of incarceration (or probation) and/or a fine and court costs, and restitution, only the last of which would benefit Sicknick's survivors. But the measure of recovery in a restitution award is much more narrow than the measure of damages in a civil action in tort (i.e. for a civil wrong).
The criminal counterpart to 42 U.S.C. § 1983 for civil liability under federal law is 18 USC § 242, but that would not apply to this case either for the same reason. The people who carried out the assault were not acting under color of law.
Immunity from state law prosecutions is discussed in another answer citing a 2003 *Yale Law Journal article (starting on page 2195) "What Kind of Immunity? Federal Officers, State Criminal Law, and the Supremacy Clause by Seth P. Waxman and Trevor W. Morrison, which is on point. But since the incident at the U.S. Capitol on January 6, 2021 took place outside of any U.S. state, the federalism and Supremacy Clause issues raised in that article which involve a different facet of immunity from prosecution rather than Presidential absolute immunity for official actions considerations, this analysis has no application to this particular fact pattern.
A doctrine called collateral estoppel also known as issue preclusion makes certain legal determinations made in one proceeding binding against the person against whom they were decided in future legal proceedings. But this does not apply to determinations of Congress regarding impeachment.
The U.S. House of Representatives, in a bipartisan majority vote, expressed the opinion that President Trump's conduct constitutes a "high crime or misdemeanor" justifying his impeachment. This week, as I write, the U.S. Senate is conducting an impeachment trial to determine if President Trump shall be convicted of the conduct for which he has been impeached and so, if he should be barred from holding future public office as a result, which requires a two-thirds majority of the Senate.
The U.S. Constitution expressly provides, however, that a finding of Congress on the question of impeachment is not binding upon the criminal courts or civil court and is limited to removal from office, dishonor and eligibility to hold public offices in the future.
If Congress purported to try to hold President Trump liable criminally for this conduct, outside the impeachment process, this would be an unconstitutional Bill of Attainder.
A civil case would also not be collateral estoppel as to the criminal case (where a different and higher burden of proof applies).
But a conviction obtained in a criminal prosecution would constitute collateral estoppel against Trump in a civil prosecution brought against Trump assuming that the civil case was brought within the applicable statute of limitations and the criminal case was resolved of the merits first. So, a criminal conviction could make it much easier to secure civil liability, albeit only through the indirect means of the collateral estoppel doctrine.
First Amendment considerations
Another answer correctly notes that on the merits of either a civil lawsuit or a criminal case, that Brandenberg v Ohio sets a high bar of First Amendment protection in incitement cases. Strictly speaking, that is an element of the underlying merits of the civil claim or criminal charges, however, and not a question of governmental immunity. One doesn't get to the merits or the First Amendment affirmative defenses at all in the analysis unless one can first overcome the governmental immunity hurdles that are presented.