The forward, who is joint top of the league's scoring charts this season with Chelsea's Cole Palmer missed FA Cup semi-final win over the Blues
A visibly frustrated Ronaldo was furious at the defensive tactics of Iceland, the smallest country at the tournament, after Tuesday's Group F clash.
"He's a fantastic footballer, but he's not a gracious human being," Arnason was quoted as saying on The Guardian's website (www.theguardian.com).
"Obviously we're not going to create as many chances as a fantastic team like Portugal but his comments are the reason why Messi is always going to be one step ahead of him. You wouldn't expect Messi to say that. It shows we got under his skin."
The defender added: "It was lovely to hear that."
Iceland take on Hungary in their next Group F game on Saturday, before ending their group stage campaign against Austria on June 22.