Can we go outside from airport in India if we have connecting flights for USA?

Let's tear this question apart because it's reasonably common. I will answer the generic question, the specific should be clear after:

1. You intend to leave the airport after stepping off a flight. If you have the legal right to stay in the country then this shouldn't be a problem. In most cases, if you are allowed inside the terminal you should be in the clear: you might simply be visa exempt or you have a visa that allows you into the country not just the airport. Citizens of countries needing Schengen airport transit visas, certain citizens who can transfer through a Canadian airport to the USA without a visa etc are exceptions. Also, if you fall under this, don't you worry, you will not be let out of the transit area. If you see a way to exit the transit area without passing a border guard / immigration officer, you are good to go.

2. You intend to reenter the secure area of the airport. You need a boarding pass (and in some countries, a government issued ID) for this. There are almost no exceptions to this rule.