No, because you created a very specific hypothetical that hinged on a failure of set-up or maintenance and not on a failure of the fixed/freewheel combination.

There is no maybe, what else do you think would happen? If you ride long enough it might start rethread but that's extremely unlikely. Once it's unthreaded itself there's pretty much no effect it can have on the bike.

Excluding mechanical brakes, there are two ways of stopping a fixed gear bike. The first is back-pressure on the pedals, the rider uses his legs to slow down the pedals and subsequently slow/stop the bike. This is often referred to as back-pressure since that's what you're doing, putting rearward (it's actually down/rear) pressure on the pedals. The second is skidding, this is where the rider stops the pedals from moving and subsequently the rear wheel, slowing/stopping is accomplished by skidding the rear tire against the road.

So it's the same, your chain gets sucked, your rear wheel will coast and you will use your brakes to slow down. Fixed or freewheel.