Diesel in F1 would be sacrilege. Really it would. These are sprint cars, endurance racing is an entirely different kettle of fish.
Whether it is feasible though, well basically the diesels would probably win, hands down. Unless some very
very careful restrictions were placed on them. They would use less volume of fuel per xyz, and without restriction, would be compound turbo'd and make insane amounts of power and torque. It would be the equivalent of the 80s turbo's vs NAs, meaning if you wanted to be even remotely competitive, you couldn't be using an NA engine.
Unlike petrol forced induction engines that begin to pre-detonate at 1 bar boost, and with ever increasing octane rating, retarding the ignition, and adding massive amounts of (wasted) quench fuel can max out at around 4-5 bars. Diesel engines are compression ignition, which means they cannot pre-detonate, and you can boost away to your hearts content, 8, 9, 10 bars of boost with compound turbos is exceptionally simple to achieve and work with, meaning that within the time it takes for the teams to adapt, Petrol would be gone from F1.
Far more relevant IMO would be bio-ethanol and other bio-mass based fuels that operate on spark ignition systems, by makeup enjoy much higher octane ratings, produce less noxious gasses and do not rely on fossil fuel deposits.