Final Settlement Dispute: How Can I Resolve Missing Salary for My Notice Period?

My final settlement amount is very low. I worked 20 days of my notice period, but they paid for only 7 days. They said my resignation request was raised on 18th Aug, and my Last Working Day (LWD) is 25th Aug, so only 7 days' salary will be payable. However, I sent an email to my manager on the 5th of Aug, and he accepted it on the same day. It is his fault as he raised the request only on the 18th of Aug. Unfortunately, he is not working in that company right now. What can be done? Please help me in this regard. I don't have a copy of my resignation email.

No, your company did wrong. You have to be paid what you have earned. You have two recourse:
1) Under the Payment of Wages Act, file a complaint with the labor department. See this for the answer here Indian Labour Laws & Shram Suvidha Simplified.
2) Under the Shops and Establishment Act, you can file a complaint with the labor department. The process is the same in both.

I do not understand what your problem is.

You worked till the 25th. They have paid you till the 25th. They would have paid the normal salary till the 18th and the notice period salary from the 19th to the 25th. If the resignation was noted on the 5th of August, your normal salary would be till the 5th and the notice period salary from the 6th to the 25th. In the end, the numbers make no difference. The total pay you get will be the same.