What happens if an immigrant marry a U.S citizen and the citizen abandon the immigrant?

Since marriage to a US citizen confers US citizenship upon the spouse, it is not clear how your spouse is an illegal immigrant unless you are also an illegal immigrant (not that you actually said that either of you are living in the US, but that seems to be the most likely interpretation). I also don't know why you are abandoning your spouse instead of getting a divorce. Nothing will necessarily happen; it depends upon what action the abandoned spouse may choose to take. If you are sued, then there could be various consequences depending upon the outcome of the suit.