The player will need healing, as his/her current HP doesn't change
The PHB (pg. 196), among other places, draws a clear distinction between characters' current HP and their maximum HP:
A creature’s current hit points (usually just called hit points) can be any number from the creature’s hit point maximum down to 0.
The passage on leveling up only refers to hit point maximum, and not current hit points, which means that characters are not healed upon leveling up.
If they are at max HP before leveling up, they are no longer at max HP afterward. For example, if a character has 20/20 HP, after leveling up they might have 20/26 HP, and would require healing to get to 26/26 HP.