The 3 pinochios comes from a WaPost article that makes a false claim attributed to Cruz:
"Cruz goes too far in asserting that the ICANN transition will result in Russia or China being able to censor U.S. Internet content."
Cruz has not claimed that. Cruz has stated that countries that officially censor content and repress journalists and free speech advocates should not have a seat at the table or a vote. Cruz then goes on to make a "slippery slope" argument, which is regarded as a valid form of argument in American jurisprudence.
Backing by google and FB for ICANN's position means absolutely nothing to the American electorate. In fact, I'd say it means less than nothing, as the top tech companies are widely viewed as tax cheats who engage in mass spying on their users and frequently violate anti-trust laws. Cruz, whose legal experience includes anti-trust work for the government and who frequently fights against bulk data collection, knows exactly how to portray their support in a negative light.