There's been recent publicity about an American diplomat in Britain
responsible for a car accident that killed someone. She claimed
diplomatic immunity and left the country.
Could the victim's family sue her in an American court?
They could sue, but the case would probably be summarily dismissed, often one of the following two grounds:
(1) related to diplomatic immunity, but not specifically covered by the Vienna Convention on Diplomatic Relations, such as common law sovereign immunity which bars suits against individuals or their employers for actions taken in an official capacity on behalf of a federal government employer (the scope of which is interpreted broadly in the case of diplomatic personnel abroad, much as it is in the case of what conduct of the President of the United States constitutes an action taken in an official capacity), unless waived. Cf. Cruikshank v. U.S., 431 F.Supp. 1355 (D. Hawaii 1977) ("Alleged activities of agents of Central Intelligence Agency in opening and photographing sealed, first-class letters mailed by plaintiff to colleagues in the Soviet Union fell within purview of general waiver of sovereign immunity statute, despite claim that Central Intelligence Agency agents involved could not have been legally authorized to carry out such activities and consequently, as matter of law, were not “acting within the scope of their office or employment,” as that phrase is used in this section.")
If a diplomat is acting in an official capacity, lawsuits can be filed against the U.S. government, but not against the individual, and can't only be brought under the Federal Tort Claims Act, and only then if an exception to it does not apply. The FTCA is the "exclusive means by which a party may sue the United States for money damages ... in tort" (28 USC § 2679. Exclusiveness of remedy). Accordingly, an FTCA action "can be brought only in a United States District Court" (28 USC § 1346(b)). Regarding the timing of filing, FTCA's § 2401(b) states that the action must be brought "within two years after the claim accrues," or "within six months after ... notice of final denial of the claim by the agency". In addition, under the FTCA, "Liability is determinable in accordance with the law of the place where the act or omission occurred" (§1346(b)(1)). More fully, 28 USC § 1346(b)(1) states:
Subject to the provisions of chapter 171 of this title, the district
courts, together with the United States District Court for the
District of the Canal Zone and the District Court of the Virgin
Islands, shall have exclusive jurisdiction of civil actions on claims
against the United States, for money damages, accruing on and after
January 1, 1945, for injury or loss of property, or personal injury or
death caused by the negligent or wrongful act or omission of any
employee of the Government while acting within the scope of his office
or employment, under circumstances where the United States, if a
private person, would be liable to the claimant in accordance with the
law of the place where the act or omission occurred.
But, 28 U.S.C. 2680(k) expressly precludes the exercise of jurisdiction under the FTCA over "[a]ny claim arising in a foreign country." See, e.g., Smith v. United States, 507 U.S. 197 (1993) (FTCA does not apply to claims arising in Antarctica). "It is a longstanding principle of American law `that legislation of Congress, unless a contrary intent appears, is meant to apply only within the territorial jurisdiction of the United States.' " Foley Bros., Inc. v. Filardo, 336 U.S. 281, 285 (1949).
Basically, sovereign immunity is absolute for torts committed abroad in an official capacity by a U.S. government official of any kind. So, if the official capacity conduct claim is barred by diplomatic immunity where it occurred, it is barred everywhere.
(2) in cases not arising from actions taken in the diplomat's official capacity, on the discretionary quasi-jurisdictional grounds of forum non conveniens, even thought American courts, as a general rule, have jurisdiction over all civil claims of private persons arising anywhere in the world against a person who is domiciled in the state where the state or federal court in question is located.
For over a quarter century, federal judges have assumed that foreign
injury cases, particularly those filed by foreign plaintiffs, are best
litigated abroad. . . . Federal judges approach the forum non
conveniens decision as if it were a species of choice of law, as
opposed to a choice of forum question.
Elizabeth T. Lear, "National Interests, Foreign Injuries, and Federal Forum Non Conveniens", 41(2) U.C.-Davis Law Review 559 (2007).
One of the leading forum non conviens cases pertinent to this fact pattern is Piper Aircraft Co. v. Reyno, 454 U.S. 235 (1981), involving an airplane crash in Scotland in which U.S. defendants were allegedly at fault. Potential loopholes in that decision have been closed by subsequent cases.
Federal judges are all that matter in a lawsuit between a foreigner and a U.S. person involving claims in excess of $75,000 (which essentially all personal injury cases worth bringing internationally do), such as the one contemplated in this case, which fall within the diversity jurisdiction of the federal courts and can be removed from a state court to a federal court. (And, state courts tend to defer to federal precedents in diversity cases in claims which are not removed from state court to federal court.)
The Alien Tort Claim Act expressly authorizes certain lawsuits in U.S. Courts for violations of international law or treaties committed abroad (although whether the judicial power extends to cases where all of the defendants are non-U.S. persons is an issue of ongoing dispute), but not for simple common law torts like the negligence actions that are the basis of most automobile accident cases.
There are also other specific statutes that might authorize lawsuits (e.g. civil rights statutes, patent laws, copyright laws, etc.), but none of them would ordinarily apply to a simple automobile accident allegedly causing a wrongful death.
It is conceivable that an attorney could find some way to thread a needle through these two strong limitations on bringing suit against U.S. diplomats in U.S. courts for personal injuries caused by ordinary common law torts committed abroad which are barred by diplomatic immunity where they occurred, but it would take extraordinary facts that do not appear to be present in this relatively routine automobile accident allegedly wrongfully causing a death of a non-U.S. person.
Could she be criminally prosecuted in America because of what she did
The Vienna Convention only directly limits criminal prosecutions of people with diplomatic immunity in jurisdictions where someone is a credentialed diplomat or head of state.
But, usually criminal cases can only be prosecuted in the jurisdiction where they are committed or the jurisdiction to which the crime was directed if a crime is committed outside a prosecuting jurisdiction but directed at a victim or target in the prosecuting jurisdiction.
There are federal statutes criminalizing conduct victimizing various U.S. government officials, but very few criminalizing conduct that would otherwise not be a crime subject to U.S. criminal prosecution if it is committed by a U.S. government official abroad (in some cases, a violation of civil rights claims might apply, but not in a car accident case like this one).
Note that I am limiting this to a criminally culpable automobile accident against a non-U.S. person while diplomatic credentials were in force. There might be a U.S. prosecution of a diplomat, for example, for espionage in the form of revealing U.S. secrets, or as another example, for a rape of one U.S. person who has diplomatic immunity by some other U.S. person at the same embassy. The cases could arguably be directed at the U.S. or a U.S. person.
The most negative U.S. consequence that could arise from criminal conduct that did not take place in the U.S. and was not directed at the U.S. or a U.S. person (for civilians not subject to the U.S. Code of Military Justice) would usually be termination of employment at the U.S. State Department and termination of diplomatic credentials, both of which would have prospective application only.