I don’t believe that’s how it works. Per this article:
[House Speaker Mike] Johnson said the resignation took him by “surprise” but that the Florida congressman did so to “start the clock” on the process for Florida Gov. Ron DeSantis to call a special election to fill the vacancy.
“So Matt would have done us a great service by making that decision, as he did, on the fly. And so we’re grateful for that so we move forward,” Johnson added.
Gaetz’s resignation will narrow the slim majority that Republicans will have in the new Congress in January, but he represents a reliably Republican district in the Florida Panhandle.
Sounds like he’s all the way out… although I suppose he could run in the special election meant to replace him. But he’d have to be re-elected all over again.