This is Turtle.

Install a new needle valve by it'self and run the water to the new
refrigerator from it. Now like chris said you should get new fitting
and tee into the line and do away with these Michey Mouse needle
valves. It would make a much better job here.

now take the needle valve that leaks through the valve and go to the
hard ware store and get you a compression cap to put on the old needle
valve and your though with fooling with it.

Now if this is a needle valve your going to use for the water to be
sent to the refrigerator as ice maker and water in the door. Whirlpool
states in there service manual that needle valves will not supply
enough water for the ice maker to work properly. They recommend that
you have a full open gate or ball valve to be at the pipe source for
the water and not a needle valve. Just a thought here.

TURTLE