Can an undocumented immigrant sue the police under 42 U.S.C. 1983 for violation of constitutional rights?

Yes.

It may seem confusing, as we often do not guarantee the same constitutional rights to non-citizens as to citizens.  However, in the United States, legal aliens may sue under 42 U.S.C. 1983, which is the statute commonly used to sue police for excessive force or unlawful arrest, unreasonable search and seizure, etc.  Graham v. Richardson, 403 U.S. 365 (1971).  Undocumented immigrants may also invoke the protections of §1983.  Phyler v. Doe, 457 U.S. 202 (1982).

If you look at the text of §1983, it states that anyone acting under color of law who deprives either citizens or any "other person under the jurisdiction" of the United States of constitutional rights are liable to the injured party.  The 14th Amendment, which extended constitutional protections to all states, does something similar in first discussing citizens and then declaring that all "persons" are entitled to due process and equal protection of the laws.

It seems our lawmakers specifically considered whether aliens would have the same rights as citizens and decided that, in at least some circumstances, they should.  I'm not sure why they decided that - and it's nearly impossible to say why Congress does anything, anyway - but there is a very good reason to extend the protection to everyone: there's no practical way to determine ahead of time who is a citizen and who is not before deciding whether to respect their constitutional rights.  Personally, I think there's another great reason to extend constitutional protections to all under our jurisdiction: it's just the right thing to do.  If rights are inalienable, they should apply to aliens.  If it is just to protect certain rights, those rights should be protected for all.