I will say Hallel at Shacharit (without a bracha) and talk with my family about the significance of the day. Yom Yerushalayim is not an exclusively Zionist holiday, although I proudly identify as a Religious Zionist. Iy should be a day for all of Klal Yisrael to commemorate our people’s salvation in 1967 from an attempted genocide, as well as the miraculous recapture of our holiest sites.
Because Yom Yerushalayim corresponds with the anniversary of an actual victory, while Yom Haatzmaut merely corresponds with the declaration of statehood, there are some who observe Yom Yerushalayim but not Yom Haatzmaut. (My family observes both.)