You describe Creeping Fig, Ficus pumila which is not an edible fruit.  It has milky sap throughout the plant.  I have never seen one quite like your picture so I will continue to research.  
Would you let me know where this plant is located?  This would narrow down my research.  Have you ever seen the fruit any color other than green.  Also, the fruit looks a little fuzzy, is this true?
Look forward to talking to you again.  A

Thank you for the further information which led me to identifying your plant/vine. You have an invasive Moth Vine, Araujia hortorum.  The fuzzy fruit is actually a seed pod.  It is a perennial climber that has escaped from residential areas and can cause major issues in wastelands and nearby forest. Check it out online.  I don't believe it you want this in your landscape and please destroy the pod.  
I have attached some pictures that may help to be sure it is your vine.  
Let me know if you have any other questions.  A

Thank you very much for your help plant/ weed will be destroyed now, appreciate the help.

You are very welcome. Contact us anytime with your questions.  A