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Yes that's how I did it.
Calculus is mostly a set of tools, analysis is the theory that's actually the math behind the computation that is calculus.
But you do want to supplement it with some intuition about continuity and measure and whatnot. Depends on your background whether you'll be able to understand.
Baby Rudin, for example, is completely devoid of the geometrical intuitions, you have to provide your own examples at every step. If you manage, then you're good to go. Spivak is calculus+analysis, may be your best bet.
Needham's visual complex analysis does rely on an understanding of calculus, but if you don't mind having supplement it with other stuff, looking up whatever is not entirely clear to you, it's also a good option, I myself read Rudin+Needham. I had already taken calculus in high school but I remembered almost nothing.
