What is the procedure of marrying a non-Muslim woman?

The requirements for a marriage between a Muslim man and a woman who follows either Christianity or Judaism are the same as for an Islamic marriage. From the Islamic point of view, a woman does not have to change her religion and convert to Islam for such a marriage to be valid. Marriage in Islam requires making a spoken offer by the bride’s father or guardian and its acceptance by the groom. This must be in the presence of a minimum of two witnesses. The witnesses are normally Muslims.
By witnesses to a marriage contract, we mean the people who are present at the time when the offer and its acceptance is made. They do not have to be official witnesses and put their names on any document. Therefore, if you invite several friends to the marriage ceremony, whether it is made in a registrar’s office, a mosque or at home, their presence is sufficient.
The other requirements are the presence of the bride’s father or guardian and agreement upon a dowry to be paid by the husband to his wife.