The phrase 'the rest of the day' is correct and usable in written English. You could use this phrase when you want to describe the remainder of a current day or when referring to the future: I plan to spend the rest of the day reading and relaxing.
The rest of the day flew by.
The rest of the day passed quickly.
And the rest of the day?
"I couldn't skate for the rest of the day".
The rest of the day she is out of luck.
For the rest of the day, the place was empty.
He spent the rest of the day sightseeing.
I love the desktop app, it’s always running on my Mac. Ludwig is the best English buddy, it answers my 100 queries per day and stays cool.
Cristina Valenza
Retail Lead Linguist @ Apple Inc.