The skipper said that conversations with disheartened followers would not change the results, while a prominent football official hit out at the on-loan Zenit St Petersburg man
Arshavin spoke to Russian fans at the team's Warsaw hotel after the 1-0 loss against Greece that eliminated them from the tournament on Saturday evening.
"The fact that we have not met your expectations is not our problem," Arshavin was cited as saying by Russian site Gazeta.ru.
"It's your problem. When I'm sitting in [Federal House] State Duma, I'll be held accountable before you.
"We lost because we did not score and the Greeks did. These conversations will not change the result of the game."
The comments have not been well-received within the country, with former Russian Football Union Leader and FIFA vice president, Vyacheslav Koloskov, stating that Arshavin is not fit to wear the captain's armband.
"Such words are not worthy of a captain," Koloskov told Championat.com.
"Arshavin has never been in the same shoes as those who, at their own expense, travel long distances to support the team.
"Recently, I met people from [the far eastern city of] Yuzhno-Sakhalinsk, who specifically came to watch good football and cheer for the national team. Many of them risk their health.
"The whole team has to bow low before these guys. The supporters come to see good football and total commitment from the players and the desire to stand up for their country.
"Instead, the team captain says it is not the players' problem, but the fans'. I believe that a captain like this should not be in the team."