Math major and mathematician can mean different things. You can be a mathematician as a hobby, but not all math majors get there. To get there, you should be actively thinking about questions in mathematics, have research interests, etc.
If you would like a career in theoretical mathematics, be advised that there aren't many jobs (where we're talking someone who was the top of his/her class from a good university would still on average have a very hard time).
You should not reconsider being a math major, however, solely based on your performance in a calculus course. It's quite possible that you will be better at the proof-based classes than some of those who did better than you in this one. Nevertheless, the advice someone gave about thinking what went wrong is a good one. You have a long way to go.
Just remember that mathematics appears all over the place, and that too many people are somewhat narrow-minded about what they believe constitutes doing mathematics. Consider the various areas (statistics, physics, engineering, computer science, etc) which can have significant interplay with mathematics before you start considering yourself on track to becoming a "mathematician."