Are there any limits on being "overly selective" during Jury Selection?

Question

Say Andy the attorney is trying to defend Nathan the Nazi in a criminal court case. During Jury Selection, where potential jurors are vetted, what is stopping Andy from saying that every Nazi-hater is not suitable to be part of the jury? And what is stopping the prosection from saying that anyone who's a Nazi sympathizer is inadmissable?

In general terms, are there any limits on an attorney's power to disqualify potential jurors until they find just the right one who is pre-disposed to side with their client?

Answer 1

One limit is that potential jurors cannot be excluded solely on the basis of their race. See Batson v. Kentucky, 476 U.S. 79 (1986); Flowers v. Mississippi, 588 U.S. ___ (2019).

As summarized in Flowers:

Equal justice under law requires a criminal trial free of racial discrimination in the jury selection process. Enforcing that constitutional principle, Batson ended the widespread practice in which prosecutors could (and often would) routinely strike all black prospective jurors in cases involving black defendants. By taking steps to eradicate racial discrimination from the jury selection process, Batson sought to protect the rights of defendants and jurors, and to enhance public confidence in the fairness of the criminal justice system. Batson immediately revolutionized the jury selection process that takes place every day in federal and state criminal courtrooms throughout the United States.

The Batson restriction is not merely that one cannot explicitly announce race as the factor by which one is excluding a juror. This will most often not be the kind of evidence available and if that were the rule, the purpose of the rule would easily be defeated. What Batson prohibits is exclusion of a juror "on the basis of race" (whether or not a proxy was the proximate or purported means by which the prosecutor made the decision). See Flowers at p. 18 of the slip opinion:

The trial judge must determine whether the prosecutor’s proffered reasons are the actual reasons, or whether the proffered reasons are pretextual and the prosecutor instead exercised peremptory strikes on the basis of race. The ultimate inquiry is whether the State was “motivated in substantial part by discriminatory intent.”

Answer 2

At least in US criminal court procedure, each side generally has a limited number of "peremptory challenges". These can be used to excuse (dismiss) potential jurors without specifying any reason, and it is very unusual for either side to specify the reason. Such challenges may not be used to exclude jurors solely on the basis of race, as the answer by Jen correctly explains in detail. It has been suggested that this limitation also be extended to other protected characteristics such as sex or ethnicity, but I am not aware of a case that has imposed such a restriction. Even the prohibition of racial exclusion can be hard to enforce, if the attorney, when questioned, asserts some other basis for a particular challenge.

Aside from the limited number of peremptory challenges, either side may "challenge for cause". This means alleging some reason why a juror would not be qualified, or would not be properly impartial. A source of possible bias would be a classic reason for a challenge for cause. But all such challenges are reviewed by the trial judge (or judge supervising jury selection, in the unusual case where this is not the trial judge), and a juror is only excluded if the judge agrees that the challenge is valid. So an attorney can suggest that people who support, or strongly oppose, Nazi parties are biased, but the Judge may well not accept this and may choose to retain such jurors. It is a judgement call.