What I think you are trying to do is use your Smart TV function for accessing the internet.
You need to press the source button so you are in TV mode, (not using the 360 input), you should be able to do this using either remote, when in you are in TV mode, (I don't have a JVC TV so don't know what buttons you have), press the Smart button you would use to access your TV's apps and internet browser like you did before, if your TV is connected to the internet it should work, is it connected by ethernet cable or wifi, your JVC remote should at least turn the TV on and off regardless of what source you are connected to, if that doesn't work the first thing I would check is the batteries as pairing the virgin remote will not disable the JVC remote.
You cannot directly access the internet using the virgin remote and it will not open up any smart function on your TV it will only control the volume and turn the TV on and off if connected by HDMI.
Hope any of this helps.