The incident occurred during half-time of the FIFA World Cup second-round clash in Belo Horizonte on Saturday, which the hosts won on penalties after a 1-1 draw.
Pinilla claims he was punched by Paiva, who insists the fracas was nothing more than pushing and shoving.
On Monday, a FIFA spokeswoman confirmed that Paiva had been suspended for one game - Brazil's quarter-final with Colombia on Friday - and that the case continued to be considered by the disciplinary committee, who could hit Paiva with a longer ban.
Pinilla took to his official Twitter account to vent his anger at both Paiva and the governing body.
"Mr Rodrigo Paiva, Brazil's press officer, assaulted me with a fist in the tunnel without reason," he wrote.
"FIFA cannot endure this offender. FIFA has to intervene on something this serious.
"(Luis) Suarez risks his career with a bite...a press officer for a fist in the face? FIFA?
"I demand from FIFA an exemplary punishment for Rodrigo Paiva as well as that of my colleague Suarez...!!! This is more serious still!"
