First, the White House did issue a readout of the call, though it was very brief. Second, the explosive claim about the tape recorder being turned off by the US side is not based on any specific knowledge on Berman's part. He's completely guessing. As he later said on Twitter, it was "conjecture." He also said he doesn't "know for a fact that they turned it off."
On top of that, no one seems to be sure whether these calls are recorded or not. In response to a question about taping procedures, a White House spokesperson told BuzzFeed News, "How these meetings are conducted is classified."
But given Berman's expertise in foreign affairs, and the fact that he works for a conservative-leaning think tank, his baseless claim quickly rocketed around liberal websites as evidence that Trump did something suspicious on his call with Putin.