Yes — but it is regulated, not automatic. An Indian resident investing in a foreign company falls under India's Overseas Investment (OI) Rules, 2022. Depending on the shareholding and control, the investment is classified as Overseas Direct Investment (ODI) or Overseas Portfolio Investment (OPI), each with its own limits and filings through an authorised dealer bank.
The structure most founders ask about — a Delaware parent that in turn owns an Indian company — is a "round-trip" structure. Since the OI Rules 2022, it is permissible for bona fide business purposes subject to conditions (including layering restrictions), which was a significant liberalisation from the earlier approval-based regime.
Practical points: the initial investment needs Form FC/ODI reporting, an annual APR filing follows, and remittances go through the Liberalised Remittance Scheme (LRS) limits for individuals. Getting the sequence wrong is hard to fix retrospectively — take advice before incorporating, not after.