I'm as big a fan of Godzilla as anyone, but as the little kid in Godzilla vs the Smog Monster / Hedorah says, "Superman beats them all."
At most, the fight against Godzilla would be the same as a fight against the Hulk, with a higher initial strength (and weight), and a nuclear firestorm breath weapon. (Plus maybe an omnidirectional nuclear pulse effect as in the second Godzilla series.) But without any possibility of the Gray Hulk's craftiness or Banner's genius affecting the fight. The end result would be the same, assuming Superman fought smart. (Comparably, Godzilla would be like a larger Abomination I guess, without the Hulk's steadily increasing strength threshold. But even if Godzilla had that...)
Which would also be true for a number of other comparable DC or even Marvel characters. Godzilla vs. the Avengers and the FF and Shield was plausible enough as the climax to the Marvel comic series, although vs. Thor was pushing it. (There are too many ways Thor could wreck Godzilla from the inside out, which could never happen with Thor vs. the Hulk. Thor didn't fight smart.) Godzilla vs the Sentinel or some similarly Superman-level Marvel character, or against various other Kryptonian (or equivalent characters), only ends in a win for Godzilla if they fight foolishly or are hampered in some other way.
Oh no: Godzilla, stuck in the crushing gravity at the center of the sun 93 million miles from Earth, is getting stronger! Whatever are we going to do?
(Also, it's ambient radiation that makes him stronger. He can clearly be damaged by focused radiant power. Lots and lots of focused radiant power on and near and inside the sun.)
Time to reinvent the oxygen destroyer? Two or three hours for Batman. Time to hurl Godzilla into the sun at close range, pushing him from an angle where Godzilla can't reach him? 20 minutes traveling at a strong fraction of lightspeed for Superman.
(Or, send Godzilla into the Phantom Zone to annoy the Kryptonian criminals there.)
The real restriction would be that Clark wouldn't want Godzilla to suffer like that, so would probably be looking to freeze him somehow for a permanent hibernation.
What's Superman gonna do with all those smarts, though? Throw him into the sun? Godzilla + Radiation = You just made things worse. Rip out his heart from the inside? Watch GMK again.
The only way to destroy Godzilla is to destroy every single cell in his body. And there's only one weapon known to do that without radiation. Now, do you think Clark is smart enough to build an oxygen destroyer?
Basically, what I'm saying here is: Batman wins, because he builds one while Supes is having a punching contest and never lets Clark live it down. Ever.
