There is nothing wrong with using the day by itself:
He has been absent since the 4th.
Of course, one would hope that context or prior knowledge within the conversation would ensure the listener knows what month is being talked about. For example, if you asked me about my Christmas travel plans, I might say:
We leave on the 22nd.
and it would be obvious that we are talking about the month of December.
You can also say:
He has been absent since the 4th of the month.
The phrase of the month is often used generically like that, either to refer to the current month, or some other month that has already been referenced, either directly or indirectly.
As for the two you ask about:
- He was absent since the 4th day.
- He was absent since the 4th of current month.
Those both sound awkward to the native ear. Just saying since the 4th day seems to ask for more information, such as since the 4th day of school, or something like that. Using "current month" can work, but I'd expect to see an article:
He has been absent since the 4th of the current month.