1. Provided the valve on the washing machine outlet works (and presumably it does if you have no machine connected) you COULD attach a line to the refrigerator from there. Note:
1a. If you wanted to also connect a washing machine, you could by using a "Y" fitting. If you do I'd strongly recommend a brass Y piece rather than plastic. Have a look at
Brass Washing Machine Y Piece 2 Way Connector 3/4" BSP | Stevenson Plumbing & Electrical Supplies - https://www.stevensonplumbing.co.uk/brass-y.html
1b. You would not need to isolate any pipework, except for using the valve on the washing machine connector.
2. The neater and more professional way to do it would be as Undertrained suggests. If your username reflects some DIY skills, you could insert a soldered or compression tee piece into the line leading to the washing machine connector, and pipe to the refrigerator from there. If you do this:
2a. You will need to be able to isolate the cold water supply to the washing machine valve. Almost certainly the main stopcock.
2b. You would be advised to fit a good quality (Pegler?) isolating valve in an accessible place in the line from the new tee to the refrigerator. It wouldn't need to be a full bore valve as there is no significant flow.
2c. Personally I'd pipe in 15mm copper as far as possible. You would presumably need some fitting to attach the refrigerator line. Whatever you use don't forget some pipe clips to keep it secure.
3. Whatever you do, do NOT use a self cutting tap. They can cause real problems.
4. If the line has sufficient pressure (well flow really) for a washing machine, it will manage a washing machine and refrigerator.