An official said 23 injured people, two in serious condition, have been rescued and taken to hospital.
The accident occurred near Kullu, a hill resort in Himachal Pradesh state, 340 miles north of New Delhi.
Rescuers found 38 bodies. Three people were feared to have been swept away by the water's strong current.
The accident's cause was not immediately known. The bus fell nearly 150 feet deep into the Beas River.
India has the world's deadliest roads, with more than 110,000 people killed annually.
COMMENT RULES: Comments that are judged to be defamatory, abusive or in bad taste are not acceptable and contributors who consistently fall below certain criteria will be permanently blacklisted. The moderator will not enter into debate with individual contributors and the moderator’s decision is final. It is Belfast Telegraph policy to close comments on court cases, tribunals and active legal investigations. We may also close comments on articles which are being targeted for abuse. Problems with commenting? email@example.com