Where are you looking in NYC, and what assumptions are you using that the calculator says rent? DH and I just bought in prime Brooklyn at the price range you're looking at and the calculator came way out in favor of buying. To make it concrete, the interest plus co-op fees on our 2BR is less than our rent was in a less nice neighborhood. Maybe you are looking at single family or 3BR apartments, but even then the rents for comparable places must be very high.
Don't forget that if you're looking at a co-op, and most places are co-ops, you will need a minimum of 1year of mortgage payments plus co-op fees in cash in the bank to satisfy the co-op board. So yes you need more than 20% down. The board will be stricter than the bank!