Can a piano play the microtones of Indian classical music?

Not exactly: a piano locks each note to equal temperament, while a raga can shade a note away from it. Bhairavi's komal gandhar sits a shade below E-flat, and Yaman's tivra madhyam a shade above F-sharp.

Pianists still play ragas within those limits. Some choose ragas that suit the fixed grid, and others let the piano's sustain carry the line.