Can an employer teminate an employee if during an employment period obtain a misdemeanor offense? (After theyve told an employee they would not fire)

upon getting hired with a nursing facility i had no criminal background whatsoever, however a month or two into employment i had a misdemeanor charge up against me, long story short i took the misdemeanor charge of smalla mount of marijuana posession. I informed the job upon my first court appearance notice, there was no problem i was told once preceedings were dealt with bring in the documents and based on what i was being charged with, i was ok with my employment and i had no worries of being terminated. two months later after the fact i told i am terminated because of my background check?