Apparently the F-117 could carry air-to-air missiles in the internal bomb bay like the F-22 and F-35 do.
https://theaviationist.com/2020/06/03/f-117s-had-an-air-to-air-capability-with-secondary-mission-to-shoot-down-soviet-awacs-former-stealth-pilot-says/
From the F-35:
The F-117 could carry a lot of different types of weapons. It would be easy enough to use a missile.
Its still a fighter verses a bomber. so apples to oranges comparison.
I think the author explains pretty well why this is unlikely, without calling the pilot's claim BS. The F-117 didn't have a radar, so could not use the radar guided missiles of that time, and opening the internal bay and somehow lowering the sidewinder enough to achieve IR lock before firing seems pretty sketchy. A Lock After Launch missile would be needed, which we didn't have yet.
After being called out, the pilot recanted his claim:
In the end, Donaldson clarified his statements on the topic and to discuss the matter in a bit more detail with The War Zone. In short, the F-117 never had the capability nor has any missile ever been fired, not even for testing. That’s perfectly understandable. Much less understandable is the reason why the pilot claimed it in the episode in first place and also confirmed it in another episode Q/A session.