The problem stems from the fact that Verizon's widespread CDMA network only supports a single session of either voice or data, but not both. This is an issue with any CDMA smartphone, not just the new iPhone.
Indeed, this issue does not exist on AT&T's network, which uses UMTS to handle both voice and data on a single session. AT&T has been touting this advantage over Verizon's network as of late, and with the new iPhone, this weakness on Verizon's side will show its ugly head even more publicly.
There was previously word that indicated Verizon would have a fix for this issue later this year, called SVDO, according to the CDMA Development Group. Unfortunately, Verizon made no mention of this during its iPhone unveiling today. There was also no indication that the Verizon iPhone would support SVDO, according to the CDMA Development Group, handsets will require hardware support for SVDO.
Our recommendation: wait for Apple's expected new iPhone announcement to happen early this summer.