Schwartzman is Jewish, and is the son of Ricardo and Silvana. He was born in, and resides in, Buenos Aires, Argentina. Because of his religion, Schwartzman was claimed by the Israeli Davis Cup team, which offered him a million dollars to represent them, but he refused it and was called by the Argentine team.
In 2010 he won the Bolivia F3 Futures (CL), in 2011 he won the Chile F14 Futures (CL). In 2012, he won titles at the Peru F2 Futures (CL), Argentina F11 Futures (CL), Argentina F14 Futures (CL), Argentina F20 Futures (CL), Argentina F21 Futures (CL), Argentina F22 Futures (CL), and Buenos Aires Challenger (CL). At the 2013 Australian Open, he lost in the final round of qualifying.