The 27-year-old finished the match, which was drawn, with seven wickets but he has been dropped from the squad for the second Test - which begins on Thursday at Edgbaston.
Robinson has apologised for the tweets, which were sent when when was a teenager, and now Dowden has questioned the ECB suspending him.
“Ollie Robinson’s tweets were offensive and wrong,” he tweeted on Monday morning.
“They are also a decade old and written by a teenager. The teenager is now a man and has rightly apologised. The ECB has gone over the top by suspending him and should think again.”
A spokesperson for prime minister Boris Johnson added later in the day: "As Oliver Dowden set out, these were comments made more than a decade ago, written by someone as a teenager, and for which they rightly apologised."