The Gorosei and the heads of the Navy know who Luffy is. Even when the rest of the world thought Smoker did everything, the Gorosei/Navy knew the truth. They even knew things that he didn't, like, the fact that his father was Monkey D. Dragon.
Garp knows that Luffy carries that same spirit in him, hence all the love fisting. He was close with Sengoku, and if he thought Luffy would do it, he definitely would of spoken up, but Luffy always told Garp "I want to be a Pirate, no Navy".
The Gorosei wouldn't want him anyway, due to the true purpose of the 7 Warlords, which is to do all the underhanded and sneaky crap the Navy can't be caught doing. They know if Luffy ever got a hold of the info they are hiding from the public, he would stop that immediately. Not that Luffy would figure anything out, but, once Nico Robin explains it to him, well you get the picture.
So, basically, he wouldn't do it and they don't want to ask him. Especially since Sakazuki is Fleet Admiral now. That dude is a whack-a-do who killed his bro.