Can a person claim medical bills for personal injury in a public place as my elderly father tripped and fell over a vacuum cleaner in a bank after turning around to get a deposit slip?

What you describe is a very typical "slip and fall" case. Yes, medical bills resulting from the fall are part of the claim for damages, but there are more elements to his damage claim beyond medical bills.

Contact an attorney in your area. They will handle your father's case on a contingency fee basis and maximize the value of the claim for your dad.