Potentially he could try, it's been done in England (though not recently, I think the last time was the bedchamber crisis and that was more an unwilling minority government) and in Belgium more recently. The advantage of parliamentary systems with figurehead leaders to manage transitions is that this can't happen (or more correctly it would be very unusual for the German President or Australian Governor General to do this, and the last time the UK monarch tried to pull this it nearly killed the monarchy).

The problem is at some point legislation for essential government functions will be required, taxation arrangements will expire, deadlines for ratifying EU legislation internally will pass etc. Belgium managed this mostly ok because while the parties didn't want to be in coalition they didn't hate each other to the point of physical violence, and the system is fully parliamentary so ad hoc deals were possible since the Head of State wont block things.

The Fifth Republic was (ostensibly, anyway) set up this way to prevent the instability of the Third/Fourth Republic and the weak coalitions and minority governments, but now they're in the same problem from the Presidential Side. They've never had this issue before, and I think the French Government does specify a Prime Minister in the constitution, unlike the UK et al where it doesn't exist but we all pretend it does, so there's an argument he's legally obligated to appoint one.