The USA has a few main priorities in its international relations: The benefit to Americans, the stability and preservation of international order (in so far as this benefits Americans). The development of peace (in so far as this maintains the international order and benefits Americans)
The US government was generally supportive of the Sri Lankan government. The Sri Lankan government was rather more pro-west than India or other South Asian countries. Moreover the LTTE (Tamil tigers) had used unconventional warfare, including suicide bombings, that had led to them being classified as a terrorist organisation. Moreover the USA was generally against separatist groups, as these tend to destabilise the international order. However, this was just one of many civil wars in the 1980s, and the USA saw no pressing need to intervene. The potential benefits (to Americans) of ending the insurgence was not worth the risks to American soldiers.
India had previously given support to the LTTE which had prolonged and intensified the conflict. America opposed this, insofar as it supported the Sri Lankan government, but again there was no pressing US interest in preventing the gun-running. So America was pleased that India was able to reach an agreement with the Sri Lankan government. The agreement was for the various Tamil groups to give up armed struggle, for the North and East of the island to be granted a degree of self-government within Sri Lanka, and for Indian troops to support as peacekeepers. The US supported this plan as it maintained the international order and developed peace. However, the LTTE was not party to this agreement and refused to surrender arms, which led quickly to India engaging militarily with the LTTE.
The attitude of America was not "we don't care what India does". It wanted India to engage in Sri Lanka in such a way that would support the Sri Lankan Government and develop peace in such a way that would benefit Americans.