"He invites us to place our trust in Him - not in trite and easy programs - but in Him and Him alone."
Despite the pressure of recent scandals, Gregory said he had already received an "affectionate and embarrassingly gracious welcome."
"The example I wish to set forth for you is that of a man filled with the faith, hope and joy of knowing Jesus Christ is in the boat."
Gregory thanked Pope Francis for the "righteous challenge - more an opportunity" to carry the Gospel message to the poor, the marginalized, and the neglected.
"Beginning today, that is my take here in the Archdiocese of Washington."
Lourdes Rivera, a local mother of 14, told CNA after the Mass that she was "overjoyed" with Gregory's homily.
"He spoke like a father - a father. I am so happy. He's our new father here in Washington and our family now feels even bigger."
Local resident Everett Jacobs added his hope that Gregory's arrival will bring fresh impetus to the new evangelization in the archdiocese.
"Bishop Gregory's pastoral spirit represents a reaffirmation of God's love for the Archdiocese of Washington. I look forward to his fresh proclamation of the gospel," Jacobs said.
Gregory's appointment was first reported on March 28, and his installation has been eagerly anticipated. Technically, Washington has been without an archbishop since Cardinal Donald Wuerl's resignation was accepted by the pope in October last year, though Wuerl himself has served as interim leader of the archdiocese since that time.
Gregory paid tribute to his predecessor during his homily, calling Wuerl a "cherished friend" and "above all, a true Christian gentleman."
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The presence of retired Cardinal Roger Mahony of Los Angeles at the Mass generated some controversy among Catholics in attendance. Mahony has been accused of impeding police investigations of clerical sexual abuse in the 1980s, and was in 2013 relieved of public and administrative duties in the Archdiocese of Los Angeles.
The Archbishop of Washington is generally viewed as one of the most influential Churchmen in the United States; the five most recent archbishops were all created cardinals - including the now-laicized Theodore McCarrick. While Gregory, 71, is widely expected to be named a cardinal in the future, it is usual for the pope to wait until the previous cardinal archbishop from the same diocese turns 80 years old and becomes ineligible to vote in a conclave. Wuerl will turn 79 in November.
As a former president of the United States Conference of Catholic Bishops from 2001 to 2004, Gregory was responsible for helping to lead the American hierarchy through the fallout of the Church's 2002 sexual abuse scandals. He oversaw the formation and implementation of the "Dallas Charter" and USCCB's "Essential Norms" in 2002.
Gregory's appointment is also historic because he is the first African-American to be appointed archbishop in Washington, D.C.; the city itself is more than 50% African-American. Gregory is also the first convert to Catholicism to be appointed Archbishop of Washington.