It's an oddball situation because normally once you are put on a heading that will intercept final, the approach clearance is included with the heading instruction ("XXX, turn right heading YYY, cleared for the approach").
But lets say the controller actually just gave you a heading to final and nothing else, and you never heard from them again. The question becomes, is this a valid comm failure or not.
If it's a real comm failure, and you continue through the final approach course and just continue off to who knows where, what is the controller to do for separation of traffic? You are now like a horse that escaped the barn, and the controller now has no idea of where you are going next.
Assuming all efforts to raise ATC fail, you go to the basic IFR comm failure procedure and proceed logically based on the comm failure protocol; clearance, expected clearance, and all that. In this case it's pretty clear to expect the approach you are being vectored to.
So if it was me, I would be frantically trying to make contact but once I was convinced I had a real comm failure I'd select 7600 and just fly the approach.
Once ATC sees the 7600 selection, that's what the controller will expect me to do and will be able to adjust traffic flow to suit. That last thing to do in a comm failure situation in a terminal area is to head off in unexpected directions.