It would be a whole different operation.

1. Choppers cannot randomly appear out of the blue. They must have come from a land base or a carrier in the vicinity. US already has these bases in Pakistan (and Afghanistan), essentially using them as a client state for more than a decade. The operation was launched from Jalalabad in the border between Pakistan & Afghanistan through mostly unguarded hills and US got to Pakistan capital in under 90 minutes. Osama operation could have been hidden as a regular reconnaissance operation. US has none of these bases close to India to launch their operation from. They cannot just move a major aircraft carrier into India's vicinity without the Indian navy knowing about it. Also, when Indians see random blackhawks flying over them (we have none of these birds at home) it is sure to gather the public & get info passed to the establishment.
2. US sold F-16 Falcons (that can intercept the Black Hawks) to Pakistan on condition that they will always be under surveillance by US Military. Thus, US will know when Pakistan is starting its scrambling jets to intercept the Blackhawks. US had little of that knowledge when it comes to Indian jets (usually bought from Russia).
3. US had extremely detailed knowledge of Pakistani working from decades of association. They even had a fake cop posted close to the site to make sure no one else come in the way.
4. US was historically an ally of Pakistan (through SEATO and CENTO) and US has been taking a lot of liberty. India has stayed neutral for the most part and it is unlikely for US to mess up that.
5. It is highly likely that Pakistan establishment was clandestinely hiding him. Thus, telling them the operation would be detrimental. For India, Osama is an enemy just as he is for the US. Thus, it is logical for US to have a joint operation with India, especially since the terrain is lot more unfamiliar for them (unlike Pakistan where they have had military operations for 3 decades).
6. US-Pakistan relationship is highly one-sided. Pissing off Pakistani leadership is not such as big deal for the US. Pakistan is not a major trade partner nor has a big geopolitical lever (allies of Pakistan are also allies of US for the most part). US-India relationship is a little more important for the US in terms of trade, technology and diplomacy.
7. There will be political trouble in the UN if US enters Indian space. In the Osama case, not many have sympathies for Pakistan. No other country protested. Everyone kind of believed that Pakistan was actively aiding Al-Qaeda.

India might not directly declare war on the US if they still chose to go on their own, but would face bigger repercussions elsewhere (including the UN). It would also be stupid as they are missing India's inputs and entering an unfamiliar terrain.

Planning & Executing the Mission to Get Bin Laden