Newcastle signed Woltemade from Stuttgart for a club-record £69m towards the end of the summer transfer window, replacing Alexander Isak.
Rummenigge, a supervisory board member at Bayern, labelled Newcastle "idiots" for meeting Stuttgart's asking price, stating Bayern would not have paid such an amount.
Woltemade has made a strong start at Newcastle, scoring two goals in three Premier League appearances since his transfer.
Stuttgart executive Alexander Wehrle responded to Rummenigge's comments by saying they were "always happy when we get congratulations from Munich."