Knowledge is rarely useless so I do not want to discourage you in any way, shape, or form from learning things that are useful to your career as a Mathematician.
For graduate admissions in the United States, you need to have demonstrated capabilities. For most Mathematics departments, a strong undergraduate Physics degree is perfectly acceptable, as the mathematical skills required for the Physics degree are very comparable to the ones in Mathematics. In the US, undergraduate Mathematics classes start with Calculus and only move towards proofs towards the end. At my university, the first course with proofs is the Discrete Mathematics class, where induction proofs are taught (among other things), which is taken at the end of the second year.
Unlike the Indian system, the Western systems allow more flexibility in switching careers. Of course, a pre-law student wanting to get into a graduate program in Mathematics might face some difficulties. You can look at the GRE subject topics to see what is expected from Mathematics graduate students. You should also look at some target universities in the US and see what they require, so that would probably involve writing to the director of graduate admissions.
In my (not often very humble) opinion, you would do best to select one of these additional classes per year or semester and do well in them. This gives you an opportunity to try out your new-found interest in Mathematics, show that you are a serious student, and might allow you to get a good letter of recommendation from outside your university. But make sure that you do not drop your grades in your Physics program.
Here is what I found from UC San Diego, which is quite good:
- A Bachelor's degree in Mathematics is recommended or a strong background in mathematics with a minimum GPA of 3.0 required.
- A year's sequence in both upper-division algebra and real analysis is strongly encouraged.
