“The Church is catholic because she has been sent out by Christ on a mission to the whole of the human race.” (CCC 831)
I again use my technique of providing answers to theological questions solely from Bible passages. (Bible verses are from the KJV.)
Did St. Paul apply “Vatican II-like” approaches to evangelism, by making his message more accessible?
1 Corinthians 9:19-23 For though I be free from all men, yet have I made myself servant unto all, that I might gain the more.  And unto the Jews I became as a Jew, that I might gain the Jews; to them that are under the law, as under the law, that I might gain them that are under the law;  To them that are without law, as without law, (being not without law to God, but under the law to Christ,) that I might gain them that are without law.  To the weak became I as weak, that I might gain the weak: I am made all things to all men, that I might by all means save some.  And this I do for the gospel's sake, that I might be partaker thereof with you.
Did Jesus deal kindly with the Samaritan woman at the well?
John 4:5-9 Then cometh he to a city of Samaria, which is called Sychar, near to the parcel of ground that Jacob gave to his son Joseph.  Now Jacob's well was there. Jesus therefore, being wearied with his journey, sat thus on the well: and it was about the sixth hour.  There cometh a woman of Samaria to draw water: Jesus saith unto her, Give me to drink.  (For his disciples were gone away unto the city to buy meat.)  Then saith the woman of Samaria unto him, How is it that thou, being a Jew, askest drink of me, which am a woman of Samaria? for the Jews have no dealings with the Samaritans.
Did Jesus state that the pagan Roman centurion who was a good man had a faith equal to those in Israel?
Luke 7:1-10 Now when he had ended all his sayings in the audience of the people, he entered into Capernaum.  And a certain centurion's servant, who was dear unto him, was sick, and ready to die.  And when he heard of Jesus, he sent unto him the elders of the Jews, beseeching him that he would come and heal his servant.  And when they came to Jesus, they besought him instantly, saying, That he was worthy for whom he should do this:  For he loveth our nation, and he hath built us a synagogue.  Then Jesus went with them. And when he was now not far from the house, the centurion sent friends to him, saying unto him, Lord, trouble not thyself: for I am not worthy that thou shouldest enter under my roof:  Wherefore neither thought I myself worthy to come unto thee: but say in a word, and my servant shall be healed.  For I also am a man set under authority, having under me soldiers, and I say unto one, Go, and he goeth; and to another, Come, and he cometh; and to my servant, Do this, and he doeth it.  When Jesus heard these things, he marvelled at him, and turned him about, and said unto the people that followed him, I say unto you, I have not found so great faith, no, not in Israel.  And they that were sent, returning to the house, found the servant whole that had been sick.
Was a Roman centurion who was not yet a Christian described as “devout” and “one that feared God”?
Acts 10:1-4 There was a certain man in Caesarea called Cornelius, a centurion of the band called the Italian band,  A devout man, and one that feared God with all his house, which gave much alms to the people, and prayed to God alway.  He saw in a vision evidently about the ninth hour of the day an angel of God coming in to him, and saying unto him, Cornelius.  And when he looked on him, he was afraid, and said, What is it, Lord? And he said unto him, Thy prayers and thine alms are come up for a memorial before God.
Did an angel approvingly communicate with a Roman centurion who was not yet a Christian?
Acts 10:5-7 And now send men to Joppa, and call for one Simon, whose surname is Peter:  He lodgeth with one Simon a tanner, whose house is by the sea side: he shall tell thee what thou oughtest to do.  And when the angel which spake unto Cornelius was departed, he called two of his household servants, and a devout soldier of them that waited on him continually;
Did a Roman centurion who was not yet a Christian have his prayers heard by God and his almsgiving honored?
Acts 10:30-33 And Cornelius said, Four days ago I was fasting until this hour; and at the ninth hour I prayed in my house, and, behold, a man stood before me in bright clothing,  And said, Cornelius, thy prayer is heard, and thine alms are had in remembrance in the sight of God.  Send therefore to Joppa, and call hither Simon, whose surname is Peter; he is lodged in the house of one Simon a tanner by the sea side: who, when he cometh, shall speak unto thee.  Immediately therefore I sent to thee; and thou hast well done that thou art come. Now therefore are we all here present before God, to hear all things that are commanded thee of God.
Did St. Peter say that those of any nation could possibly attain acceptance with God and salvation?
Acts 10:34-35 Then Peter opened his mouth, and said, Of a truth I perceive that God is no respecter of persons:  But in every nation he that feareth him, and worketh righteousness, is accepted with him.