He said the Catholic Church had not dealt with abusive priests decisively or quickly enough and said its top priority now was helping victims heal and regaining their trust in the church.
The Pope's comments, to reporters on board his plane, marked his most thorough admission to date of failings in the way the sex abuse scandal was handled.
He also spoke of protests planned for his UK visit, saying Britain had a "great history of anti-Catholicism. But it is also a country with a great history of tolerance".
He said he expected a warm welcome from Catholics and other believers and "mutual respect and tolerance" among those with anti-Catholic sentiments.
"I go forward with much courage and joy," he said.
Asked about polls that suggest the faithful had lost trust in the church as a result of the sex scandals, he said he was shocked and saddened by the scope of the abuse, in part because priests take vows to be Christ's voice upon ordination.
He said he felt "sadness also that the church authority was not sufficiently vigilant and not sufficiently quick and decisive to take the necessary measures" to stop the abuse and prevent it occurring again.
The Pope arrived at Edinburgh Airport aboard Alitalia flight Shepherd 1 at 10.30am to begin the first papal state visit to the UK.
He was welcomed by the Duke of Edinburgh and later met the Queen at the Palace of Holyroodhouse.
The Pope was officially welcomed by the Queen with a ceremony full of pageantry at the start of a tour meant to strengthen ties with the UK's six million Catholics and the wider nation.
The highlight of the visit for Catholics will be the Pope's beatification of Cardinal John Henry Newman - the 19th century cleric who converted from the Church of England.
But the trip comes amid renewed anger at the worldwide child abuse scandal that has engulfed the Roman Catholic Church and dogged the Pope's own religious career.
That anger has gathered pace with recent revelations in Belgium of hundreds of new victims, at least 13 of whom committed suicide over the years.
Victims abused by priests, and secular campaigners, have called on the pontiff to go further than an apology and hand over all information about suspected abusers within the church.
In his comments on board his plane, the Pope said abusive priests must never have access to children, saying they suffered from an illness that mere "goodwill" could not cure.
The Pope is widely expected to meet some of those who suffered during his visit to Britain.
In another development, German-born Cardinal Walter Kasper, 77, withdrew from the Pope's entourage after suggesting Britain was like a "Third World country" with its multi-cultural population.
Vatican officials attributed the change of plans to ill health and distanced the Church from the comments.
But the Cardinal was under growing pressure to apologise for his comments.
It is the first papal trip to Britain since Pope John Paul II made a pastoral visit in 1982 following an invitation from the Church.
Thousand lined the streets as the Pope was driven through Edinburgh, and past the controversial Scottish Parliament building, to the Palace of Holyroodhouse for his meeting with the Queen.
Addressing the Queen and assembled guests at the Palace of Holyroodhouse, the Pope said: "Thank you for your gracious invitation to make an official visit to the United Kingdom and for your warm words of greeting on behalf of the British people.
"In thanking Your Majesty, allow me to extend my own greetings to all the people of the United Kingdom and to hold out a hand of friendship to each one.
"It is a great pleasure for me to start my journey by saluting the members of the Royal Family, thanking in particular His Royal Highness the Duke of Edinburgh for his kind welcome to me at Edinburgh Airport.
"I express my gratitude to Your Majesty's present and previous Governments and to all those who worked with them to make this occasion possible, including Lord Patten and former Secretary of State Murphy."
The Pope acknowledged First Minister Alex Salmond and members of the Scottish Parliament.
He said: "As I begin my visit to the United Kingdom in Scotland's historic capital city, I greet in a special way First Minister Salmond and the representatives of the Scottish Parliament.
"Just like the Welsh and Northern Ireland assemblies, may the Scottish Parliament grow to be an expression of the fine traditions and distinct culture of the Scots, and strive to serve their best interests in a spirit of solidarity and concern for the common good."
The Pope paid tribute to the work of Christians in UK history, including Florence Nightingale and John Henry Newman, whose beatification he will celebrate on Sunday.
He described Cardinal Newman as "one of many British Christians of his age whose goodness, eloquence and action were a credit to their countrymen and women".
"These, and many people like them, were inspired by a deep faith born and nurtured in these islands," he said.