My grown nephew shorted out my receptacle in the utility room, so I called my niece who recommended her friend, a licensed electrician to fix the problem.

While he was able to get power on quickly, he said the box was too small and there were too many wires and they should be put in a bigger box. I agreed that it should be done, but he said he wouldn't do that work unless he could combine my GAS dryer to work on the same circuit as my washer. I don't believe this needs to be done because the only other items working on the dryer line are two receptacles, one with a GFI for my Gas cooking stove for the igniter and another that I rarely use. Also my washer only has one other receptacle on it's circuit that's used for a table lamp.

I don't think either of these circuits is over loaded as it stands and I don't see the need to pay to have a dedicated circuit put in to combine the laundry room appliances.

We can only second guess what happened. How did the outlet get shorted? Why does the stove have a GFI? I could see his logic in not wanting to overload a circuit or have too many wires in the box. We don't know what caused the short or how it was wired. Boxes are rated for the number of wires in the box.

but he said he wouldn't do that work unless he could combine my GAS dryer to work on the same circuit as my washer.
That's not logical. Ideally the washer would be on a circuit by its self and the dryer would be on as circuit by its self. Both supply motors which have a heavy current draw, especially at start up.
That being said as table lamp on the washer circuit certainly would not be a problem. Same thing for the stove igniter on the dryer circuit.

You certainly would not put the washer and the dryer on the same circuit. Perhaps you misunderstood and he wants to run a separate circuit for the washer and a separate circuit for the dryer. Which is not necessary if you haven't had any problems tripping breakers.

I don't see how the shorted outlet and the box it is in, is relative the washer and dryer wiring unless the circuits for the washer and dryer run through that box. That would mean that the outlet is on one of those circuits.

Is the shorted outlet used for the washer or the dryer?