It is correct. However, 'profuse apologies' is so excessive that it sounds almost like a joke, at least in British English. We might use it between friends in a light-hearted way when we are deliberately being over-formal. I am aware that Indian Business English has its own conventions, though, so it may be appropriate in that context. For an international context, it would be more normal to offer your 'sincere apologies'.

It is grammatically correct. It would be understood. The truth is, an apology is an apology. How effective will it be? It depends on many things. One is the fact that you are apologizing. Another is whether you are accurate in saying what you are apologizing for. Another is what you promise to do in the future. The exact form of words is much less important. I do not know your culture. "Profuse" does not sound like a good choice unless it is a standard and customary formula. In the United States it is not. "Profuse apologies" literally means "many, many, many apologies" and usually carries a sense of "too many." It would normally be used in describing somebody else. It would normally be used when someone apologizes over and over and over again--often, empty apologies. To strengthen the word "apologies," the customary words in the United States would be "sincere apologies," or "heartfelt apologies" if the situation involves bad behavior and hurt feelings. Here is an example of the use of the phrase "profuse apologies" was used in a real magazine article. Notice that it is describing somebody else's apologies, and that the apologies--despite being "profuse"--were rejected: "Many years ago, when our kids were little, Kevin was driving carpool and showed up late to pick up the neighbor’s children. It wasn’t the first time. After Kevin's profuse apology, the children’s dad was rather matter-of-fact. He said, 'Don’t be sorry; change.'" In the United States, this would be a good apology in a business situation: "Our sincere apologies for the mistake we made on your bill. We have made a refund, which should appear on your credit card within three business days, and we are mailing you a coupon for a 20% discount on your next purchase." Notice that word "sincere" isn't even needed. The rest of the message shows sincerity by clearly stating the problem, acknowledging whose mistake it was, and stating what will be done.

Hey Ritika, it is grammatically correct to say in a formal letter. Another option would be to "offer my sincere apologies". Hope this helps!