Fixing the connection on ACCOR PUSHON® R-SERIES Supply Stop Valve
The "tube that connects the connector and the pushon valve" in your picture looks like soft clear vinyl tubing; if it is, that was the cause of the failure. 

Your valve has a compression type connection:

 

and is designed for tubing (copper or poly/plastic) that is fairly rigid rather than soft. The connection utilizes a ferrule:



which I do not see any sign or mention of in your pictures or question. You will likely need to replace that short piece of tubing with a section of tubing that is compatible with both the connector and the valve's compression fitting, like poly tubing: