It is known that eastern priests can be married. On the other hand Western rite claims priest to be unmarried. So consider Uniate eastern catholic priest who is married. Can he, if he wants can he say a Latin Rite Mass?
Priests and bishops of the Eastern Churches in full communion with the Bishop of Rome may concelebrate at any celebration of the Eucharist in any rite, including the Latin Rite, provided they have the permission of the diocesan bishop or eparch.
The Code of Canon Law is silent on this issue, but the Code of Canons of Oriental Churches (CCEO) says,
A concelebration between bishops and presbyters of different Churches sui iuris [of which the Latin Church is one] for a just cause, especially that of fostering charity, and for the sake of manifesting unity between the Churches, can be done with the permission of the eparchial bishop, while observing all the prescriptions of the liturgical books of the principal celebrant, having removed any liturgical syncretism and wearing the appropriate vestments and insignia of his own Church sui iuris (Can. 701).
Except for concelebration, as mentioned, priests and bishops may not simply celebrate the Eucharist using a different rite from their own, unless they receive an indult from the Holy See. This is commonly called becoming “bi-ritual.” (The Syro-Malabarese eparchy of Adelaide, Australia, made a good summary of how to deal with the pastoral situation of members of an Eastern Church living in a predominantly Latin-Catholic area.)
In practice, it is quite common for Eastern priests to receive the indult to celebrate in the Latin Rite (but not vice-versa, probably because the Latin Rite is generally simpler to learn, and because the Latin Church is so pervasive), and married priests from an Eastern Church are in no way impeded from receiving it.
So in summary: any Eastern priest may concelebrate in the Latin Rite, with permission from the diocesan bishop, and likewise, any Eastern priest may solicit from the Holy See an indult to celebrate in the Latin Rite as the principal celebrant (just as a Latin-Church priest may do, in order to celebrate an Eastern Divine Liturgy).
Whether that priest is married is not a consideration.
It is rare but under certain circumstances a Latin Rite priest may be married. Usually this occurs when a Priest/Minister from Lutheran or Episcopal church petitions Rome (local Bishop) to become Catholic and at the same time remain/become ordained as a Latin Rite priest. This usually occurs due to great upset in a church - such as German Lutheran pastors petitioning after WWII. They would take all the vows except that of celibacy and would not be allowed to re-marry if their spouse should pass away. It is not prohibited just not the norm.