Question: Do I need to remove my watch before putting on my tefillin?
Answer: The Mishna (Megilla 24b) writes that there mustn’t be anything in between one’s tefillin and their arm (chatzitza). Thus, the Shulchan Aruch (OC 27:4) writes that one should ensure to place one’s tefillin directly on their head and arm. The Rema, based on the Rashba (1:827), qualifies this to the tefillin boxes, though allows one to have something under the straps (See Magen Avraham OC 27:5).
The Mishna Berura (27:16) writes that the leniency only applies to the winding around the arm (kerichos). There mustn’t be any chatzitza by the tying (keshira), however (See Shaar Hatziyun 27:16). Similarly, the Aruch Hashulchan (OC 27:13) writes that the minhag is to be particular not to have a chatzitza even with the arm straps.
R’ Ephraim Greenblatt (Rivevos Ephraim 4:16:7) writes that according to the Chasam Sofer (YD 192) one must remove one’s rings before wrapping tefillin. The windings round the fingers are considered keshira and so a ring would be a chatzitza.
R’ Ovadia Yosef (Yechave Daas 3:2, Yabia Omer 2:OC:2) writes that especially if one wears one’s watch normally at the end of their arm after the seven wrappings, it doesn’t act as a chatzitza.
Nonetheless, he writes that while we shouldn’t prevent others from wearing their watch, it is ideal to remove it.
In conclusion, one should remove one’s ring before wrapping one’s tefillin round that finger. While one is yotze if they keep their watch on, it is commendable to remove it.