Red Rocks? No way.
There's an interesting comparison to be made with Eldo.
Overall, I would say the climbing is not that similar between The Gunks and Eldo, but... Many many routes in Eldo have moves on them that, if you changed the rock from red to white would seem exactly like Gunks moves. Particularly on the many direct west-facing routes, where the strata is often just like at the Gunks. You get a good horizontal that takes gear just below a roof. Follow a weakness through the roof, and find positive holds at the lip. Pull hard, rock up over, find gear, and relax.
And the rock, though sandstone rather than quartzite, was baked hard enough in a volcanic zone that parts of it are nearly as compact as quartzite. It's halfway to being metamorphosed into a quartzite conglomerate. I could probably name a dozen routes off the top of my head that have moves identical to moves at the Gunks somewhere on them.
With that said, the overall experience of climbing on Eldo rock does *not* feel like at the Gunks. The different types of cracks, the overall more broken feel, the different environment etc.