I think it's pretty safe to say Gai can but just chooses not to. There are a number of scenes of Lee's history that pretty much elaborated how Lee is special (or hopeless) because he can't use ninjutsu. If Gai, a jounin, couldn't do it, I don't think there would be anything for Lee to worry about, having such a model. Besides, one only needs to have his name on the scroll, not carry it around or even possess it. After all, Naruto's name is on the scroll Jiraya carries around (for who knows what purposes - to gather more frog summoners?).

Perhaps Gai just uses little ninjutsu for his own reasons and lately because he wants to keep the bond between him and Lee strong. As long as he relies on taijutsu only Lee can purely look up to him.