The UEFA ban rules Demiral out of his team's quarter-final against the Netherlands on Saturday, and the semi-final, should Turkey progress.
The Turkish Football Federation joined Turkish government officials in denouncing the suspension but said they cannot appeal against it because it is under the three-game threshold.
After scoring his second goal in Turkey's round-of-16 win over Austria, Demiral made a sign with each hand that is associated with the Turkish ultra-nationalist organisation Ulku Ocaklari, wh oare more widely known as the Gray Wolves.
Demiral said it was an innocent expression of national pride and that he was hoping he would have "more opportunities to do the same gesture again".
But it was condemned as "racism" by German interior minister Nancy Faeser, and Cem Özdemir, a German politician of Turkish descent, said the gesture "stands for terror, fascism."
UEFA said they banned Demiral "for failing to comply with the general principles of conduct, for violating the basic rules of decent conduct, for using sports events for manifestations of a non-sporting nature and for bringing the sport of football into disrepute".
Turkish soccer federation president Mehmet Buyukeksi denounced the decision as being politically motivated and accused the European soccer body of "double standards".
"When compared to the fines and suspended penalties for much more serious offences, including racist behavior in the stands, this two-match ban is hugely disproportionate," Hurriyet newspaper quoted Buyukeksi as saying.
Sports minister Osman Askin Bak said the "unfair and biased decision ... has no legal basis and is entirely political".
Turkey's Foreign Ministry said the decision "has reinforced the view that there is an increase in the tendency to act with prejudice against foreigners in certain European countries".
Meanwhile England midfielder Jude Bellingham has been cleared to play in the Euro 2024 quarter-final against Switzerland after being given a suspended one-match ban for a gesture he made in the last-16 match against Slovakia.
The Real Madrid star gestured towards the Slovakia bench after scoring an overhead kick to force extra time.
Bellingham insisted in a social media post that the gesture was "an inside joke towards some close friends", but UEFA opened an investigation on Monday to determine whether the gesture "violated the basic rules of decent conduct".
UEFA announced on Friday that Bellingham had been fined $48,200 and issued with a suspended one-match ban. The ban will be triggered by a further offence within one year.
