If I was fired from my job because I was accused of falsifying a medical record, can I collect unemployment?

It depends on whether the accusation is true or not.  If the accusatioin is false, you would be entitled to receive unemployment benefits.  If your employer is unable to prove that the accusation is true, you should prevail on your claim.  It is possible that you may have to appeal the initial determination if the ruling is not in your favor.  This will give you the opportunity to testify before an appeal referree who will make a decision based on the testimony and evidence presented at the telephone hearing.

No, in this case you most likely are ineligible for unemployment insurance. Falsifying a medical record would be grounds for termination for cause--it can be seen as variously violating company policy, committing fraud, or even committing a crime in some circumstances. When someone is fired for cause, they may not receive unemployment compensation.