I would like to know if an Orthodox man marries a Catholic woman, and this were his second marriage, would the Church recognize this marriage?
In order to give a most complete answer, I would need more information than that which you have provided, such as whether or not the Roman Catholic party had ever been married, whether the Orthodox man’s first marriage ended through the death of his spouse or through divorce, whether the Orthodox man’s first marriage was to an Orthodox Christian in an Orthodox ceremony or whether it was to a non-Orthodox individual in an Orthodox ceremony or a non-Orthodox ceremony, etc.
Based on the information you provided, and according to the best of my knowledge, here is a response to your enquiry:
If an Orthodox man who had been married and then widowed marries a Roman Catholic in an Orthodox service, it would be recognized by Roman Catholicism and, of course, it would also be recognized by the Orthodox Church.
If an Orthodox man who had been married and then widowed marries a Roman Catholic in a Roman Catholic service, it would be recognized by Roman Catholicism but it would not be recognized by the Orthodox Church, inasmuch as an Orthodox Christian should only engage in marriage in an Orthodox ceremony.
If an Orthodox man who had been married and then divorced marries a Roman Catholic in an Orthodox service, it would of course be recognized by the Orthodox Church [provided all requirements for such situations in the particular Orthodox jurisdiction in which the ceremony takes place have been satisfied in advance] but it may not necessarily be recognized by the Roman Catholic Church, which may require an order of annullment by a Roman Catholic marriage tribunal of the Orthodox man’s former marriage. [To my knowledge, depending on the exact circumstances of the former marriage, this may or may not be the case.]
If an Orthodox man who had been married and then divorced marries a Roman Catholic in a Roman Catholic service, it would of course be recognized by Roman Catholicism [in fact, it would not be celebrated unless the requirements set forth by Roman Catholicism for divorced individuals, i.e., requiring them to obtain an annulment] but not by the Orthodox Church for the same reason enuerated in point 2 above.
There may be numerous other variations on this depending on the exact circumstances, the practices of the Roman Catholic diocese with which one is dealing, the requirements for obtaining a blessing for a second marriage in specific Orthodox jurisdictions, etc.