plasticplumbingvalve
I tried to connect the needle valve to plastic tubing, but the tube slides out if I pull on it. More over the metal no longer fits over the tube because it has already been tightened. 
Since one side no longer fits I tightened the other side to stop the water for now but for sure this will not hold. What is the right way to go about this?
I'm not sure who the idiot was who put this down that hole, but it was pretty firmly lodged in there. I cut off a bit more of the PVC and was able to get my trusty pry bar in the notch and turn it. After about three quarters of a turn the stupid thing came out. And sure enough, there was a regular stop and waste down there underneath whatever the heck this is.
The blue plastic piece itself can't be replaced. The whole inlet valve needs to be replaced. Pretty sure it's this part which costs about $50. Not very familiar with taking apart dishwashers down that far but the appliances I have taken apart have been surprisingly easy. I've always found videos online that helped. This video looks like your dishwasher. https://www.youtube.com/watch?v=MNMd_lDdG_w looks very easy... but then again so does screwing in a supply hose. :)
Considering it's a new appliance and your inexperience with repairing dishwashers you may want to call Whirlpool. It may even be covered under warranty.
Very surprised to see that it's made of plastic.
Best Answer
Your assembly looks right. The tubing could be nicked, though, so when you try again snip off an inch from the end of the tubing so you'll have a fresh surface to work with. Also verify the mating face of the valve is clean and undamaged.
One common cause of leaks with poly tubing (at least it used to be for me) with the use of a brass ferrule is that the ferrule can get misaligned a little too much. It gets compressed a little out of round and won't seal. It's more prone to happening with poly tubing because poly is softer than copper tubing. The job definitely can be done with a brass ferrule; I'm only saying it can be a little challenging and don't feel badly that it has taken a few tries.
You could try the connection with a Delrin (aka acetal or "plastic") ferrule as illustrated in this answer. They're a bit easier to work with on poly tubing. These ferrules are generally available alongside the other fittings for tubing in a plumbing department.