Can I run an adult site in India if I have a server in the United States?

If you are running a legal adult in the US UU., Then there will be no problems of EE. UU :RE and from the Indian point the laws are confusing.

Are there any restrictions on the registration of domains in India?

The Government of India, some years ago, established the Computer Emergency Response Team, or CERT-IN, as a means to monitor all incoming and outgoing Internet traffic from India. Currently, there is no established law as to which websites the government censures, or when. In the past, websites, mainly pornographic and anti-establishment, have been blocked.

Adult websites are restricted for being against public morals and decency. Most registrars and hosts will not register or host websites that contain adult content, such as pornography, warez sites, etc. However, the discretion about what is "adult" material has been left entirely to the registrars and hosts of the websites.

Therefore, the best thing is if you want to run a site for adults with privacy protection for domains.