Barron Hilton -- Criminal Investigation CLOSED .... Unless ...

The criminal investigation into the brutal beatdown of Paris' Hilton's brother is on ice ... because Barron Hilton is not cooperating ... TMZ has learned.

TMZ broke the story ... Lindsay Lohan's buddy, Ray LeMoine, smashed in Barron's face at an all-night rager in Miami. Barron says Lindsay instigated it, but Ray says Hilton was the aggressor.

Barron immediately filed a police report after the Dec 6 incident, but he has not cooperated with cops ... who have asked for a meeting. In fact, we're told he hasn't even answered the phone.

So the Florida State Attorney's Office tells us ... they are shelving the investigation and will ONLY fire it up again if Barron cooperates.