Is fuel required for a satellite to orbit the Earth?

No. Many satellites have been launched without engines or any means of station keeping.

Or yes, but only because they are slowed by friction from the thin traces of upper atmosphere that satellites pass through (and some other minor influences), so they occasionally need a boost. Otherwise they would orbit forever without need of propulsion.

Satellites in low orbits require more boosting than ones in higher orbits. (There’s more air down low.)

The Earth has orbited the sun for billions of years without needing propulsion. There’s little or no solar atmosphere this far out to cause friction. Same with the moon — it’s far enough from Earth that there’s almost no atmospheric friction.

Note that you don’t specifically need fuel, or engines — you just need some way of boosting the velocity & altitude of the satellite. That’s probably engines, but it could be something else.