The 57-year-old bishop succeeds Cardinal Antonios Naguib, who officially retired last week. The cardinal had a stroke in 2011.
The Holy Father said that Sidrak's election to the patriarchal throne is "an important event for the entire Church."
"I am sure," wrote the Pontiff, "that, with the power of Christ, victor over evil and death by His resurrection, and with the cooperation of the fathers of your patriarchal synod, in communion with the college of bishops, you will have the courage to guide the Coptic Church. … May the Lord help you in your ministry as 'Father and Head,' to proclaim the Word of God, so that it may be lived and celebrated with piety according to the ancient spiritual and liturgical traditions of the Coptic Church and may all the faithful find comfort in the paternal care of their new patriarch."
Christians make up 9% of Egypt's population. There are no more than 250,000 Catholics out of a total population of 83 million. There are some 8 million Coptic Orthodox in Egypt, making them the largest Christian minority in the Middle East.