Good advice given above, don't argue on the frequency, sort it out on telephone later.
Reminds me to an incident many moons ago, when PanAmerican was still flying and I was an active controller. Area control, the sector south of the airport. The Clipper took off, off-peak TMA did not need him and I got him directly on my freq after take off in northerly direction. I saw nothing turning south for quick identification on the radar screen, so squawk ident. Still nothing where the aircraft was expected to be, only a thick ident of an aircraft heading north. Could not be my one, or could it? Asking to squawk ident once more, the pilot replied : ehr, control, I'm sorry, afraid we made a mistake of 180°, turning south to the SID now if you approve, hope we didn't put you in too many difficulties.
Quick coordination with APP as well as north and east sectors, as the aircraft was rapidly climbing. Luckily we could all take it with a relieved smile, in particular thanks to the attitude of the pilots admitting their mistake before I as the controller would be tempted to rap them over their knuckles on the freq. Even a tel con after the flight wasn't necessary anymore let alone an official report.
I realise it wouldn't work in all circumstances, certainly not when it is about misinterpretations, but this incident may show how to prevent/stop a nasty exchange of words on overloaded frequencies.