My husband and I had an argument and now he has to go for fingerprints but I don’t have any complaint and don’t want to press charges. Can he cross the US border?

This is not an uncommon event, when a fight breaks out between two people and the police are called, they can press charges for assault even if you do not wish this to happen.  Your husband will have to get prints done and go to court but hopefully the assault charges can be withdrawn in court especially considering that you do not wish to press charges.  Since he will be fingerprinted, he will need a file destruction to remove the prints so that he can safely cross the US border without worry of being denied.