When I was in high school (in the early 1960's), Euclidean geometry was the only course in the standard curriculum that required us to write proofs. These proofs, however, were in a very rigid format, with statements on the left side of the page and a reason for every statement on the right side. So I fear that many students got an inaccurate idea of what proofs are really like. They also got the idea that proofs are only for geometry; subsequent courses (in the regular curriculum, not honors courses) didn't involve proofs. The textbook that we used also had some defects concerning proofs. For example, Theorem 1 was word-for-word identical with Postulate 19; Theorem 1 was given a proof that didn't involve Postulate 19, so, in effect, we were shown that Postulate 19 is redundant, but the redundancy was never mentioned, and I still don't know why a redundant postulate was included in the first place. Another defect of the standard courses in geometry was that, because of the need to gently teach how to find and write proofs (in that rigid format), very little interesting geometry was taught; the class was mostly proving trivialities. I was fortunate to be in an honors class, with an excellent instructor who showed us some really interesting things (like the theorems of Ceva and Menelaus), but most students at my school had no such advantage.

I conjecture that Euclidean geometry can be used for a good introduction to mathematical proof, but, as the preceding paragraph shows, there are many things that can go wrong. (There are other things that can go wrong too. I mentioned that I had an excellent teacher. But my school also had math teachers who knew very little about proofs or about geometry beyond what was in the textbook.) So my advice is, if you want to develop a course such as you described in the question, proceed, but be very careful.

Incidentally, many years ago, I recommended to my university department that we use a course on projective geometry as an "introduction to proof" course. The idea was that there are fairly easy proofs, and the results are not as obvious, intuitively, as equally easy results of Euclidean geometry. My suggestion was not adopted.

Qiaochu Yuan's suggestion of discrete math instead of geometry might have similar advantages as my projective geometry proposal, but it will still be subject to many of the pitfalls that I indicated above, plus one more: Most high school math teachers know less about discrete math than they do about geometry.