Yes. If you ignore the tidal effect and relativity and any change in mass (planets radiate light and lose atmosphere and add space dust and meteors all the time, so mass isn't constant), then in a two body system with no outside effects, the orbit would remain perfectly circular. There would be no outside force to affect the circular orbit. A circular orbit is impossible because nothing can be that exact, but on a computer simulation you could set it up and it would remain circular.
For your scenario to work you'd need to give both the planet and moon infinite hardness, so they wouldn't bend at all and fixed mass and space would need to be completely empty of anything else. Needless to say that's impossible. But only in Newtonian gravity.
Relativity creates a very very tiny decay in orbits, in your system of a planet/moon that would be close to negligible but there would be a very tiny spiraling inward. The relativistic effect on an orbit was first noticed with Mercury's orbit around the sun (and Mercury isn't falling into the sun, it was noticed by other effects - but lets not get into that here).
Similarly, any loss in mass, gain in mass or orbital drag (because space is full of tiny particles, fast moving particles, photons and neutrinos, all of which cause a tiny but at least in simulation, calculatable drag), then the two body system would have an imperceptibly small spiral and not be a perfect circle. You could say in a sense that it becomes elliptical but it would be more like a constant very tiny force where, once it was elliptical, it could wonder back to more circular. Not all perturbations or drag on an orbit make that orbit more elliptical, it can work in either direction.
It's worth noting that "falling" or decaying towards the planet wouldn't "create" an elliptical orbit. A circle is an ellipse. You asked specifically about 2 body systems, where, ignoring tides, falling in or out would be more of a slow spiral. An ellipse isn't the result of a decaying or perturbed orbit. An ellipse is the baseline orbit. Perturbations and orbital decay happen on top of the ellipse (if that makes sense), they don't cause the ellipse.
In a 3 or more body system you get orbital perturbations on the orbits. Those often remain stable, they're just variations that mostly move back and forth. See Eccentricity variation and Apsidal Precession.