Anyone can file a lawsuit at any time. The question is, what is the odds of success? Non-greencard holders who are refused visas for the US, or refused entry to the US, have a very low likelihood of success, given which most immigration attorneys will advise that (absent some special circumstances) filing such lawsuits is a waste of time.
(If you can make the case that the visa/entry refusal was due to some improper reason, such as racial or religious discrimination, political vendetta, government corruption, etc., then you might have some chance, but even then the odds are not that great. But if your case is simply "they won't let me in due to my prior felony conviction in a US federal court but I don't think that's fair", then your odds of success are almost exactly zero.)
(If you can make the case that the visa/entry refusal was due to some improper reason, such as racial or religious discrimination, political vendetta, government corruption, etc., then you might have some chance, but even then the odds are not that great. But if your case is simply "they won't let me in due to my prior felony conviction in a US federal court but I don't think that's fair", then your odds of success are almost exactly zero.)
(Disclaimer: IANAL.)