He is wrong.
You’ll need to look up resources for your state.
Is the repair due to damage that occurred while you were living in the dwelling?
Basically, normal wear is not considered damage, but actual damage is, such as scraping away ice in an old freezer, and puncturing the freon tube, or whatever, even accidently.
Aging paint and worn carpeting is not damage.
Exactly how is he “trying” to make you pay?
If it’s just a letter, ignore it.
Threaten to notify the “powers that be” of the out of code things...
Is he withholding your last paycheck?
Second, you should have a labor contract that specifies the housing. In that doc, you should have the info you need. If it specifies you are responsible for certain maintenance or damages you are. Second link is an example contract. See para 13.
If you want to fight it you may need to go to small claims court. See your courts website for instructions.
http://www.mossadams.com/articles/2014/october/employer-provided-housing
http://fels.net/Data/Laws/Housing/Housing-Sample-Agreement-Eng-Spn.pdf