Are there tires that grip on gravel? I think you mostly just need to be careful there. If the gravel is extensive enough that you'd need an aggressive tread, the road bike may be a problem unless it's an old road bike that will take a decent size tire (32-ish). If it's just a little bit of gravel that strays onto otherwise good pavement, I wouldn't let that determine my tire choice.

When you say 'desert' I hear 'goathead'. Is puncture protection a priority? If you have goatheads, the Schwalbe Marathon Plus is the obvious recommendation, but it's awfully heavy for a hilly route. For a non-goathead level of puncture protection, I like the Continental Gatorskins or Grand Prix 4 Seasons.