Currently - No there is not an option.
You could replace the dim text with opaque text but that can get very messy and double the work.
LD
Dimensions are completely controlled by Revit.. I have attached an image for reference.. You can control the read conversion but vertical is not in there..
So you are pretty stuck on this one.
LD
10 years later... how is this STILL not an option?!
Autodesk - help us out! -- Add 'Vertical' to 'Dim Text Reading' orientation.