During a power outage I attached my generator incorrectly (I have never done this before - I just was not paying attention). Anyway, I crossed the ground at the box with the black. When I turned on the power and then the specific circuits inside the nearest GFCI arced and smoked. I turned off the power immediately and found my error. The rest of the house is working fine now on generator power but the one GFCI outlet will now trip every time I turn on that specific circuit which feeds a run in the basement. I did replace this outlet with a new equal amperage GFCI. Did I damage the specific breaker? As I said, everything else is working properly and no trips or issues - just this breaker. It is immediately adjacent to the board in the basement.
You probably overheated and burned every component in that line, including the wiring and the GCFI outlet.
You probably melted/decomposed the insulation on that circuit. The GFCI is tripping because you have significant leakage on the hot or neutral lines, which means they're leaking to either the ground or to some other path within the house.
You may have damaged wiring on other circuits in the house, even if they appear operational.
I don't think it's possible to burn out a breaker unless it was already defective. It interrupts the hot leg and has no grounding, so it has nowhere to short to, and will shut off it senses too much current, saving itself and whatever is on that circuit.
Circuit breakers have a maximum current that they're rated for, usually around 10,000 amps for a residential one. You can look it up for your model if you're worried, but it's probably way, way more than your generator is capable of generating.
In fact, the circuit breaker is probably the only electrical thing in the whole house that you can pretty well guarantee was not damaged. Anything else is suspect.
I agree with others: your wiring may well have been damaged. I would say you should shut the power off to as many circuits as you can bear, particularly the GFCI one, and call an electrician immediately. Even if a circuit is functioning OK right now there may be burnt/melted insulation or damaged connections just waiting to start a fire. I assume the electrician will want to do an inspection of all the circuits, similar to what you would do after a lightning strike where they electrically test the insulation and connectivity properties of the circuits.
Clearly something is still wrong, so you should not just brush this off as a near miss.