Yes, Indian law allows the registration of marriage between an Indian and a foreigner.
Such marriages are usually registered under the Special Marriage Act, 1954, which governs civil marriages irrespective of religion or nationality.

Legal Framework:
Special Marriage Act, 1954
Most common law used for registering marriages between an Indian citizen and a foreign national.
Allows interfaith and international marriages.
Requires a 30-day notice to the Marriage Officer.
Parties must reside in India for a certain period before registration.

Conditions for Marriage (Section 4 of the Act):
Either party must reside in India for at least 30 days before giving notice.
Both must be:
Legally competent (age: groom 21+, bride 18+)
Mentally sound
Not already married (unless lawfully divorced or widowed)
The marriage must be consensual.
Procedure to Register a Marriage with a Foreigner:
Step 1: File Notice of Intended Marriage
File at the Marriage Registrar’s office in the district where either party has resided for at least 30 days.
Form: Form I under Rule 5 of the Special Marriage Rules.
Step 2: Notice Publication
The registrar displays the notice publicly for 30 days to allow objections.
Step 3: No Objection Certificate (NOC)
The foreigner must obtain a NOC or affidavit from their embassy stating they are unmarried and legally free to marry.
Step 4: Documents Required
Passport and visa of the foreigner
Proof of age and address for both parties
Photographs
Passport-sized photos
NOC (for a foreigner)
Affidavit regarding marital status and nationality
Step 5: Registration and Solemnization
After 30 days, if no objections are raised, the marriage can be solemnised in front of 3 witnesses.
A marriage certificate is issued.

What if the Marriage Took Place Abroad?
If the marriage is performed outside India, it can be registered under the Foreign Marriage Act, 1969 through the Indian consulate or embassy, provided the formalities were followed.