What size is an ice maker connection?
There are different standards to pipes and threads. Your issue is that you are trying to mix two different threaded standards.

The ball valve that you linked to has national pipe threads (NPT). NPT are tapered and it is that taper that makes the seal between a male and female joint. No washers are required.

What is shown in your photo is compression fittings which have a straight or running thread. The seal is made by either a ferrule or a rubber washer.

In your case, the connection between the copper tube and the supply hose has been made as follows:

1/4" Copper Tube > 1/4" Compression by 1/4" Compression Coupling > 1/4" Supply Hose

The seal on the copper tube is made by a ferrule that is located under the nut. As the nut is tightened, the ferrule is compressed between the coupling and the pipe.

The seal on the supply hose side is made by a rubber washer which is usually built-in the the nut of the supply hose. (The compression coupling would have come with two nuts, one of which has been discarded as the supply hose has one built-in.)

So, to answer your question, you simply need a 1/4" compression x 1/4" compression valve that will replace the current coupling. Here is a link to such a valve.