Yes they can.

After the liberalization in 1991, foreign companies can have 100% FDI in most sectors of the Indian economy. The license Raj which required all investors to partner with the Indian Government or an Indian national was removed them.

Only a certain number of critical sectors like Defense and so on have a maximum FDI limit. Even that has been stretched to 50% recently. So yes, anyone can come and start a company in India and provided they are not in the extremely small number of critical sectors like Defense, they can have 100% ownership without having an Indian citizenship.

India is a fast growing economy with over 50% of the population being under the age of 35 and the level of literacy going up rapidly (currently at around 75% with the number being higher for the younger generation). It is a good place to start a business but it will also be tough due to the extreme competition from other young entrepreneurs. The Government is also aiding new startups with various plans like the Startup Village, Make in India and more. NGO’s and certain individuals and organizations provide help for startups as well.

So if you’re interested, it is a good place and time to start, though be warned - it will take a lot of time and effort to succeed. Expect 100+ hours workweeks for atleast 10–15 years before your business is fully settled and brings in the big money especially if you’re not going into the software sector.