If someone steals my car and the police refuse to respond, and in fact prevent me from taking action to recover the car.
Can police be sued for not doing their job? I.e., if I spend millions of dollars on a breach of trust case, will I win?
The police have no affirmative legally enforceable duty in the U.S., to the general public, or any particular person who invokes their assistance, to take action to enforce the law.
The leading case on point is Town of Castle Rock v. Gonzales, 545 U.S. 748 (2005) (full text here). In this case, despite the urgent requests of the beneficiary of a Colorado domestic violence restraining order and the warning that a violent crime was imminent, the police in the Town of Castle Rock blew the woman off, and did nothing to prevent ex-husband who had illegally abducted her three daughters from murdering them even though they could easily have done so. The conclusion of the majority opinion by Justice Scalia states that (some citations and footnotes omitted):
We conclude, therefore, that respondent did not, for purposes of the Due Process Clause, have a property interest in police enforcement of the restraining order against her husband. It is accordingly unnecessary to address the Court of Appeals’ determination that the town’s custom or policy prevented the police from giving her due process when they deprived her of that alleged interest.
In light of today’s decision . . . the benefit that a third party may receive from having someone else arrested for a crime generally does not trigger protections under the Due Process Clause, neither in its procedural nor in its “substantive” manifestations. This result reflects our continuing reluctance to treat the Fourteenth Amendment as “ ‘a font of tort law,’” . . . but it does not mean States are powerless to provide victims with personally enforceable remedies. Although the framers of the Fourteenth Amendment and the Civil Rights Act of 1871, 17 Stat. 13 (the original source of §1983), did not create a system by which police departments are generally held financially accountable for crimes that better policing might have prevented, the people of Colorado are free to craft such a system under state law.
Despite this concluding statement, there was a very solid argument that the law of the State of Colorado at the time had already been intended to do just that, but was not and would not be given effect.
This U.S. Supreme Court case dealt with claims under federal law.
There is no affirmative constitutional right to reclaim a stolen car or try to prevent your car from being stolen by a private individual, even if you have a legal right to do so under state law. So, you can't bring a federal civil rights action to enforce a deprivation of that right.
The situation is even worse under state law, where police have absolute immunity from liability for all tort claims (i.e. roughly speaking, lawsuits to enforce rights other than contractual rights), unless it comes within a specific exception (of which are there are several) for cases of ordinary negligence such as car accidents when just driving around and not having emergency lights illuminated, or failing to clear ice from a sidewalk on police department property.
Their boss may not take kindly to their indifference, but that is a purely discretionary issue for their boss in the context of an employer-employee relationship (often further limited by civil service protections for public employees and/or protections as members of a law enforcement union).
The primary exception, not applicable in the example in the question, is that law enforcement officers have certain obligations to protect people who are in law enforcement custody from harm. This obligation is stronger prior to a criminal conviction, and is weaker after someone has been convicted of a crime and is serving a sentence of incarceration for that crime.
If you brought a lawsuit of the type described in the question, you would probably lose the case on a motion to dismiss filed swiftly after you filed your lawsuit, and there is a very good chance that you would be compelled to pay the legal fees of the people whom you sued.
Comparative Law Footnote
Incidentally, while this rule of law is widely shared in countries with "common law" legal systems derived from English law, it is not the case with nearly the same degree of clarity in "civil law" legal systems based upon the civil codes found in Continental Europe such as France, Germany, Spain and Italy. Almost all countries in Latin America, most countries that are not former British colonies in Africa, and most countries in East Asia and Southeast Asia, are civil law countries.
In civil law countries there generally are both statutory obligations and human rights law obligations for which the government or government officials can have liability for willfully refusing to enforce criminal laws. But even there it is, as a practical matter, difficult to enforce this obligation and is possible only in the most stark circumstances (such as the one described in the question where there is open defiance of this obligation without any understandable and legitimate contravening reason for failing to act).
A class action lawsuit has been filed against the City of Seattle, surrounding the de-policing of the infamous occupied zone. This overlaps Castle Rock v. Gonzales in part, and can be distinguished because a person has an uncontroversial property right to their property under the US and Washington Constitutions. This is different from Castle Rock, where plaintiff did not have a property right to police protection. The allegation in the suit is that the defendants aided the infringement of their property rights, thus subjecting citizens to deprivation of rights. I am not particularly sanguine about the prospects for this case, but they have at least pointed to a basis for a different ruling.
The two issues that I think will be most prominent in this case are the fact of an actual property right, and the question of government action. For example, if the police participate in a lynching by handing a person over to a mob and they provide the rope and transportation to a tree, their actions clearly violate the person's federal rights. There are plenty of successful cases of this type. If the police stand by passively and let a mob seize a person and hang them, their deliberate indifference is probably a cause for legal action (I can't point to a specific case right now so that is a conjecture). In the CHOP lawsuit, the defense will undoubtedly argue that their actions (of withdrawing the police from the precinct, giving the park to the occupiers, providing concrete barriers to the occupiers) are within reasonable discretionary scope of the government. The distinction between action and inaction is in some views spurious (deciding to do and deciding to not do are are both actions), but in light of Castle Rock, I think the courts will not be too amenable to that perspective. They may however take notice of the actual direct actions of the city.
I honestly don't see how you could argue that police action directly caused the violation of your property rights (I'll take this to be a hypothetical, not an actual case). If you attempted to defend your property and they physically restrained you to facilitate the theft, you could have a case; or if the police encouraged the thieves by announcing that they would not respond to 911 calls. If you later found your car and they prevented you from repossessing it, that could be within the scope of proper police action (it's the courts, and not the police, who judge which party is the rightful owner of property). I think it comes down to the question of what it is that the police did do.
If police violate some law other than one requiring them to do their job and also don't do their job, you can sue them for violating that other law.
For example, accepting a bribe is probably actionable under § 1983 if the acceptance of the bribe causes you harm.
But law enforcement officers have no legally enforceable duty to members of the public to do their job as explained at the post: Do police have a civil duty to do their job? to which the answer is "no" outside of special circumstances like limited duties not to act with deliberate indifference to harm occurring owed to people who are already in custody.
Many things that seem like they should be obvious in the law, aren't true.
Similarly, for example, it is not well established legally that you are entitled to be released from prison if you are lawfully convicted of a crime in a constitutionally compliant hearing process, even if you can prove later in a habeas corpus petition that you are factually innocent of the crime.
Another example is that you have no constitutional right to an appeal of a criminal conviction (this right is purely statutory), although there are constitutional rules governing how a right to appeal must operate if a statute or court rule gives you a right to appeal a criminal conviction.
The distinction between a law enforcement officer's duties and a marriage clerk's duties is basically because the authority of a law enforcement officer to take actions to stop crimes is a discretionary duty, while the duty of a marriage clerk to issue marriage licenses which you reference is a non-discretionary duty.
Usually, someone who relies upon a government official's performance of a non-discretionary duty does have some form of legal relief available (although not always a lawsuit for money damages).
The following anecdote describes the realities and practicalities of attempting to obligate the police to do their job.
A driver in the US was stopped by police for illegally making a left turn. As the officer was writing the ticket, the driver saw another car making the same illegal left turn, and asked the officer why he didn't stop that car too. The officer replied, "That's not how it works."
In other words, the police cannot practically enforce every law for everybody every time.