The doctor asked me to have a record of food routine like what I eat in/on one day. Which is grammatically right to use?
I'd use "in", because it aligns with other uses of time measurements. "How many eggs can you eat in five minutes?" "How much water can you drink in an hour?" The word refers to a time interval. The sentence "How many pages of a book can I read on a day?" doesn't make as much sense as "in a day".
The word "on" when used with a date is short for "falls on" or "is recorded on". "Your wedding anniversary falls on a Friday this year." "I was born on the first day of the year." It is a point in time, not an interval.