Silk armor was used for this purpose by several real historical armies. It's not going to STOP the blow the way I think you're imagining, but it does a few very useful things. It cushions the blow quite a bit, and against piercing weapons the silk will bind up around the point. So, to your point, it DOES get forced into the person wearing it, but that's actually one of the key benefits.
In the case of arrows it makes them much easier to remove without causing further damage, AND by keeping a layer of material that has its own antibacterial properties between your innards and whatever you got stabbed with, it dramatically reduces the chances of infection.
So, if you want armor that will completely protect the wearer from getting slashed or stabbed, silk isn't going to do that. You need something very rigid and strong like plate armor for that.
BUT, if you want very lightweight, comfortable armor that doesn't slow you down, and makes it much more likely that you're going to SURVIVE being slashed or stabbed, silk will do that all day long and twice on Thors-day.
EDIT: I feel like I need to address a misconception I'm seeing in a number of responses, and emphasize the counterargument others have raised. If your indestructible silk has the same flexibility as real silk does, and you're wearing it loose, it will not prevent sharp force trauma. I have personal experience with this. It was denim in my case, not silk, but a sharp steel edge opened up a gash four inches across in my leg all the way down to the bone without actually cutting through the denim. You'd need to have padding underneath to prevent the silk from just forming an edge around the blade. Otherwise you still get wounded, it's just a very clean wound.
EDIT the Second: I didn't REALLY address indestructible silk versus the normal kind very well, so I'll elaborate a bit. Assuming that indestructible silk (like mithril) is rare and expensive, and if you have some you want to use it as effectively as possible, there are a couple ways you'd do it.
If you're a wealthy urban noble and you're worried about assassins and duels, then you can use it all by itself under more normal clothing, with all the pros and cons I noted above. You would ESPECIALLY want to make sure you've got matching GLOVES, because being able to protect your hands and arms from getting sliced during a rapier duel or knife fight is a HUGE advantage. The general wisdom in knife fighting is just accept that your arms and hands are going to get cut all to hell, and to keep your opponent from hitting anything vital until you can beat him, and hope you don't die of blood loss first. Rapiers are sharp, but don't have a lot of mass behind them so against a weapon like that, the silk will often cause what would be a deep stab into a glancing poke.
Now, if we're thinking about battlefield armor where you're expecting to get hit, and hit HARD, all day long, then you'd want the silk to be part of a gambeson or suchlike, with the silk layered over nice thick padding everywhere except the joints. If it were me, I might want a tough layer of leather over the top of that in certain places for things like fire, because silk doesn't really prevent heat transfer all that well.
All that would only slow you down about as much as a really heavy wool coat would (I used to have a Surplus Swedish Army Greatcoat I was very fond of), but would allow you to at least survive just about any human-scale armed strike I can think of, and would shrug off entirely anything that wasn't a direct hit.
