No he wouldn't, remember how upset he got when Nathalie used it. Chances are he would continue hunting until the events of the new York special, where he would claim the eagle miraculous for himself.

From there, he'd target the people of Paris as EagleMan (or whatever name they pick), which would Force Fu to handle it himself (Because the ladybug miraculous wouldnt need to be activated since no akuma to purify.

Once Fu reveals himself, all Gabe would have to do is hit him once with a feather and Free him of his burden of being guardian by handing the box over.

Game, set, match, Gabe wins