India decriminalised homosexuality when it scrapped a colonial-era ban on gay sex in 2018, but it remains a taboo topic in this socially conservative country of 1.4 billion.
Prime Minister Narendra Modi's government has in the past refused to legalise same-sex marriage.
The RSS, established in 1925, is estimated to have millions of active members across India and overseas.
Lawyers for the couples declined to comment about Bhagwat's comments, saying the matter was before the court.
Reporting by Rupam Jain, Arpan Chaturvedi; Editing by Miral Fahmy and Raju GopalakrishnanOur Standards: The Thomson Reuters Trust Principles.