row names are names so they are always character.
I recommend you dont use them anyway as they tend to confuse more than you would benefit from them, the benefits could be seen with lower cost from making them a regular column containing the same information.
The row names indicate dates. I could transpose my data so that they become the names of the columns. Can I then change the class to integer?
The reason I need to do this is because I'm working with a financial time series and all the dates became numbers (the number of days since 1899-30-12). I want to change them back to dates but in order to do so, I need them to be integers and not characters.