Well, whatever “support” it has or doesn’t have among whoever, it has support in the Code of Canon Law.
ALL clerics (deacons, priests, bishops), married or unmarried, are bound to perfect, perpetual continence. (There are many married deacons, some married priests, and no married bishops.)
That is certainly true, in the Roman Rite of the Catholic Church. However it is not true in the Eastern Rites. Dogmatizing the issue would necessitate the suppression of the immemorial custom and discipline of the Christian East and coercing them into kowtowing, yet again, to the Latin Church. You might want to consider the legacy of your Bishop John Ireland who many Orthodox jokingly refer to as the founding father of the Orthodox Church in America.
After reading this article, I could not figure out when this man became a priest, but in this picture his wife has a baby in her arms.
Of course, we can’t ever really know, but I find it hard to believe that this priest is continent. Then again, this could be yet another example of how the post-Vatican II church turns a blind eye to true Catholic teaching (assuming what you have stated is true teaching: continent married priests).
http://www.cruxnow.com/church/2014/08/05/im-a-catholic-priest-and-im-married/