Is it legal to build an app that is built on top of another company's software?

Possibly. The "very popular social media iOS/Android app" itself will be protected under copyright and, potentially, patent(s). Depending on how your software will be interfacing, through the third-party API or otherwise, with the app may raise Stored Communications Act, Computer Fraud and Abuse Act, and privacy-related issues. If a problem does develop, the fact that you were using the third-party API will not likely prevent the app owner's claims from being at least asserted against you. A good answer to your questions will be very fact-dependent and it is thus very advisable to consult with local counsel in private before substantial development.