With 4 days you could include GC and Vegas, but it would be a bit of a 'whirlwind' tour. From Denver head for Moab for the night [Arches NP?] and take 191 south and through Monument valley on 163 to 160 to 89 and then south to 64 that will take you into the GC south rim NP via the East entrance. Spend a night at GC and continue to Vegas for night 3 and then continue to LA. If you could spare another day [or two] you would have some time to explore some of the places that you will be driving through.

As mentioned, weather could play it's part and cause delays, so you may as well have your options open and go with the weather and let each day unfold.