Connecting an Ice Maker hose directly to a tap
You need an adapter that will transition from your tap's thread to a compression thread. 

Most (U.S.) refrigerators are equipped with 1/4" O.D. tubing which utilize 1/4" compression fittings. Your picture seems to show something larger than 1/4" O.D., my first guess would be 3/8" but if you are not in the U.S. I guess it could even be a metric size.

As for the tap size, I think it could be garden hose thread (adapter shown in top picture), or pipe thread (adapter shown in lower picture).

You will need to head to a good plumbing shop and describe your problem, bring along photos which include your measuring tape for scale.