I was told that if Mom falls one more time, the LAW requires her to move out. Is that true?

It probably depends on the Assistant Living facility.   Example, my late Dad lived in Independent Living but he was becoming more of a fall risk plus his memory was failing.

The Staff recommended he move to their Assisted Living/Memory Care building.   To which he did move.   The Staff was more than happy to help him with his falls and any other medical issues.   And Dad got more physical therapy to help him keep his balance.   One issue for Dad's falls was a bad knee to which at 95 he didn't want to risk surgery.   And Dad was good about using his rolling walker.

Read the Florida health department regulations. They do require the facility to evaluate and move patients who no longer meet evaluation.

I think that means her level of care has changed and she is no longer considered to be able to live in an ALF, they won't assume that responsibility and will request you move her to a skilled nursing facility. Assisted living is just that; if the person cannot meet their standard I believe they have the right to ask the person to move to a NH. If you mom has fallen several times it doesn't sound like she can live independently in an ALF - she sounds like she needs more supervision. ALF do not portend to provide 24 hr supervision for a resident - that is clear. If your mom's care level changes she may need 24/7 care. This is common.