Problem #1: the yes in "Yes, we can" has an assertive character to it in American English. (I'm guessing the si in your Spanish example has a similar gusto to it in Mexican.) はい ("hai", yes) is your standard affirmative. It's like a normal English "yes" and doesn't really carry the oomph which the "yes" in "Yes, we can" carries. Japanese already has its own grammatical terms which fulfill this role, e.g. よ or ぞ as sentence-ending particles, or other conjunctions which can signify that the speaker is sternly and robustly informing the critical listener that he can do whatever it is that he's set out to do.
Problem #2: the we in "Yes, we can" makes it clear that the speaker is familiarizing himself with us and saying that we (and possibly others, too) can do whatever it is he's alluding to. That's totally absent from はい、出来ます. The sentence might mean, "Yes, I can." Or it might mean, "Yes, we can." Or it may even mean "Yes, he can." All it literally says (in English) is the word "yes" (はい) and the present tense form of the verb "to be able to do" or "[to] can do" (出来る). Translation: "Yes, can do." Is it I can do? We can do? He can do? ^^; It's unclear. And in Japanese, the necessary means by which to render the case more clear are going to make the slogan feel awkward and unnatural.
Problem #3: when one translates, he strives to translate meanings, not words. This is because communication itself, both oral and written, is simply a means to an end: and the end is conveying and receiving ideas from yourself to your audience and from your audience to you. I think you may get caught up in this tricky matter if you try to literally translate "yes, we can" into Japanese. While it may be grammatically correct, it doesn't feel like something I would expect to hear at a Japanese rally. I would expect to hear things like やるぜ! or 一緒、やれ!or some such. Sayings whose verb tenses don't make much sense when translated back into English but which feel right at home in Japanese.
But then again, I'm not Japanese, either; and I'm not fluent, either. So take what I've said with a grain of salt, too, I suppose.