I have a Schengen multiple entry visa, valid only from 24th June to 4th July, applied through the German Embassy. I will be in Frankfurt from 24th June till 1st July, after which I plan to fly to Rome for a 3-day trip. I will arrive in FRA airport from Rome on 4th July and then travel back to the U.S on the same day. (I am an Indian national residing in the U.S)

Will I have problems clearing passport control since 4th of July is the last day of my visa validity?

You shouldn't, unless your return journey is delayed for some unforeseen reasons. You will go through passport control in Frankfurt as you move from Schengen to non-Schengen part of the airport.

Why would you go through passport control? If you are flying in and out of Frankfurt Airport on the 4th you should stay airside - just don;t go landside at all - then you don't get involved with immigration at all.

What Alec is saying is there will be no passport check when you arrive in Frankfurt from Rome since both are Schengen zone.

The only passport check will be when you depart FRA.

The OP is flying in from Italy on July 4th, which is an inter-Schengen flight with no passport check. But in order to catch his flight back to the US, he/she has to go from Schengen to non-Schengen part of the airport, and there is a passport control, exiting the Schengen area.
But as OP still has a valid Schengen visa, shouldn't encounter any problems.

nyt, you don't get involved in immigration at all from Rome to FRA, so there's no issue with roaming freely about the terminal or even leaving the airport if time permits.