What is an appropriate way to do it to an Indian person?
In my (albeit Dutch) experience, the appropriate way to handle this isn't going to be any different just because your coworker grew up in India. You say you're not particularly close, so treating this with a more professional than personal attitude would be fine. I've worked with people from several cultural backgrounds now, and unless you're extremely close, say to the point where they've invited you to participate in their cultural practices, just being Dutch/professional has always worked fine for me in situations that required the offering of some support in a workplace environment: it ends up being appreciated.
My main guideline for situations where work obligations and private life 'milestones' overlap like this is that things need to be consistent: Any appropriate way of handling these situations treats everyone as equal and avoids people feeling bypassed/underwhelmed when it's their turn to receive sympathies or congratulations. This often means adapting to already existing practices within a team, or proposing to start a new practice within a team. It also means that what is or isn't appropriate in your specific situation can probably only be decided by you, but I hope the above guideline can help you do that.
I was going to also ask if some gesture (like sending a card/flowers/...) is appropriate and appreciated, but I would have to find his home address for that (possible).
Some teams I've been on sent cards to congratulate/sympathize all the time, others just reply to e-mails/whatsapp messages. I personally like the approach of the first kind better, as it is more likely to reach people that aren't checking their work laptop/e-mail/phone in such circumstances.
With Covid-19 and the increase in remote work, I also see a bit of an increase in situations that a card is sent for: Where you'd usually be bringing and sharing treats with your coworkers to celebrate your birthday, and get your congratulations while in the office, they now send you a birthday card and a chocolate bar you don't have to share.
Based on that, I think sending a sympathy card wouldn't be out of line here at all. It's often done in the places where I work, though usually on behalf of the entire team and not on a personal account. So that's something you might want to consider/check: if someone is sending a card on behalf of the team, or if the team wants to start a practice of doing so.
I want to add that I and colleagues intend to be as flexible as possible with his deadlines.
You can most certainly send him an e-mail that expresses your condolences regardless of whether you send a card or not, but only start about the deadlines if you actually have control over those, so you won't be making empty promises or raising false expectations. If you do have that kind of control, I'm sure that assuring him that you'll be as flexible as possible with the deadlines, and offering to work together to sort this situation out will be appreciated.
The few times I caught a flu and was ill for a week, it was nice hearing that deadlines could be moved or coworkers could pick up the slack. I can imagine the same feeling applies in situations where you're dealing with dead family members.