Are the drops of water resulting from prostatitis after urinating pure (i.e., not ritually impure) secretions?

This statement is incorrect. The basic principle is that everything that exits from the private parts is impure, with the exception of semen due to evidence of its purity. These drops may be prostatic fluid (wady), which is impure by scholarly consensus, and it is obligatory to wash whatever it has touched on the body and clothes. It also necessitates ablution (wudu').