Yes, a person holding tourist visa in India can get married in that duration in India. It is advised that you get your marriage registration in India . Unfortunately there is no specific spouse visa in India. However, you can get your Tourist Visa converted into an X visa, by a time consuming process without leaving the country.
Prior to September 2012, all applications for the extension and conversion of tourist visas on the grounds of marriage had to be directly made through the Ministry of Home Affairs (MHA) in Delhi. Now, the task of processing applications has been delegated to Foreigner Regional Registration Offices (FRRO) and Foreigner Registrations Offices (FRO) across India. This means that instead of going to Delhi for an interview, you'll need to apply at your local FRRO/FRO.
The main documents required for Tourist to X Visa conversions are:
Recent photo in the specified format.
Passport and visa.
Spouse's Indian identification (such as Indian passport).
Proof of residence.
An Indemnity Bond on 100 rupee stamp paper, signed by the spouse.
Report from the relevant local police station about marital status, including observations, confirmation of living together, and security clearance.