If an American citizen marries an illegal Mexican immigrant and records the license in his county is the marriage considered to be legitimate?

Yes. It is not necessary to record the marriage in Mexico for it to be considered legitimate. For Immigration purposes, any marriage which meets all of the following criteria is legitimate:

1) The marriage was legal in the place where it happened.

2) All other prior marriages were legally terminated before the marriage took place.

3) The two parties were of the opposite sex.

4) Both parties were physically present at the ceremony, unless longstanding cultural or religious traditions forbid such.

5) The marriage has been consumated. (The couple has had sex at least once since the marriage.