How long would it take for a small cut underneath the eye to fully heal And will it leave a scar?

If the cut is clean-edged and not infected, it should heal quickly and leave no scar. Timing depends on the depth of the wound in this case, but as you describe it was "small" I'd say it'd be mostly gone in 1-2 weeks and invisible in a month.

A "rip", burst laceration, etc. will be more inclined to scar as the edges rarely meet up exactly. Keeping it sterile is key here.