Absolutely NOT in June. Delhi experiences scorching heat in June. While I have survived many Junes in Delhi, as I grew up there, and have spent a few more on since I left the U.S. Those who have time and can afford, head to the mountains, to places like Mussuorie (once a nice retreat, but now a filthy tourist ghetto. Nearby Dhanulti is very nice and peaceful), Nanital, Simla, and Kashmir etc. Some embark upon pilgrims to Badrinath, Gangotri etc. Blood bank donations drop as even those who stay do not venture out much unless they have to.

However, I did take my wife too India in June. And she did visit the Taj Mahal in June. the white marble looked dirty, you could not stand barefoot on it. My only two visits to the Taj have been in June. The vist to the Taj is the best after the monsoons, especially then first full moon night after the monsoons.