Today is the Feast of the Nativity of St. John the Baptist. John is one of the few saints to have more than one feast day on the liturgical calendar (his martyrdom is celebrated on August 29), and perhaps the only saint besides the Blessed Virgin to have his birth celebrated as a feast day. That means that we receive a surprising number of questions about John here in Catholic Answers' apologetics department.
Why is June 24 celebrated as John the Baptist's birthday?
Am I right in thinking that the date of June 24 for the Feast of the Birth of St. John the Baptist was based on Luke 1:26? There we find the phrase "in the sixth month," which I suppose links the previous events of the conception of John with the Annunciation event.
It is more likely that St. John the Baptist's feast day was determined after the Church chose the date on which to celebrate Christ's birth. June 24 is almost exactly six months before December 25, chosen by the Church to be the Nativity of Christ. Once the date for Christmas was chosen, related feasts could then be placed on the calendar based on Scripture and Tradition.
It is also worth noting that other feast days on the Church calendar are placed in similar fashion. The Annunciation is on March 25, nine months before Christ's Nativity. The Immaculate Conception of the Blessed Virgin is on December 8; nine months later, we celebrate the Virgin's Nativity on September 8. For a long time, the circumcision of Christ was commemorated on January 1, eight days after Christmas when his circumcision would have been performed according to Jewish law. (Mary's Motherhood of God is now commemorated on January 1.)
Why is John's birthday celebrated anyway?
It is more usual for saints to be commemorated on the date of their death, which is in fact the day of their birth into eternal life. But John, like Christ and the Blessed Virgin, has a feast day for his earthly birth. His birth was important in itself because it heralded the birth of Christ, but there was another reason that set John apart from other saints.
What exactly is the status of this belief that John was born without sin. Is this a doctrine of the Church or what?
The belief that John the Baptist was freed from sin in his mother's womb is a pious sentiment that is taken from the scriptural account of the Visitation in Luke 1:41–44 when the Holy Spirit indwells Elizabeth and prompts her baby to recognize the unborn Christ Mary is carrying. It is not an official teaching or doctrine of the Church.
Is John really the Prophet Elijah?
This is one of the most asked questions about John, mainly because of John's denial that he was Elijah, and Jesus' affirmation that he was.
In the first passage you quoted, John was being questioned by the Jewish religious leaders about whether he was Elijah. Their question was whether he was literally Elijah returned from heaven. In answer to that question, John said "No" because he was not a literal reincarnation of Elijah.
In the second passage, Jesus is answering a question of the disciples about the fulfillment of prophecy. The disciples want to know why the scribes say that Elijah must return before the messianic age. Jesus answers that he has returned and the disciples understand him to be referring to John the Baptist. In this case, Jesus is speaking metaphorically. John the Baptist is not literally Elijah, but he has acted as Elijah in heralding the arrival of the Messiah.
St. John the Baptist in today's world
As noted above, St. John the Baptist's martyrdom is celebrated on August 29. The Gospel of Mark tells us the story of his death:
For Herod had sent and seized John, and bound him in prison for the sake of Herodias, his brother Philip's wife; because he had married her. For John said to Herod, "It is not lawful for you to have your brother's wife." And Herodias had a grudge against him, and wanted to kill him. But she could not, for Herod feared John, knowing that he was a righteous and holy man, and kept him safe. When he heard him, he was much perplexed; and yet he heard him gladly.
But an opportunity came when Herod on his birthday gave a banquet for his courtiers and officers and the leading men of Galilee. For when Herodias's daughter came in and danced, she pleased Herod and his guests; and the king said to the girl, "Ask me for whatever you wish, and I will grant it." And he vowed to her, "Whatever you ask me, I will give you, even half of my kingdom." And she went out, and said to her mother, "What shall I ask?" And she said, "The head of John the Baptizer." And she came in immediately with haste to the king, and asked, saying, "I want you to give me at once the head of John the Baptist on a platter."
And the king was exceedingly sorry; but because of his oaths and his guests he did not want to break his word to her. And immediately the king sent a soldier of the guard and gave orders to bring his head. He went and beheaded him in the prison, and brought his head on a platter, and gave it to the girl; and the girl gave it to her mother.
When his disciples heard of it, they came and took his body, and laid it in a tomb (Mark 6:17–29).
I do not think it is coincidence that two years ago, on the feast of St. John the Baptist, the so-called Marriage Equality Act was passed by the Senate of New York State. I do not think it is coincidence that this year on St. John the Baptist's feast day we are again waiting a verdict on the sanctity of marriage, this time from the U.S. Supreme Court.
St. John's defense of marriage in his day took the form of telling a king that his attempted marriage was unlawful. It earned John the enmity and wrath of a queen who could not stand to be told she was not really married. Even John's imprisonment could not satisfy her, likely because the injustice of his imprisonment was evident to all. His very presence in chains was a rebuke to Herodias. Rather than amend her life, or seek to persuade the king to amend his, instead Herodias decided to erase John's living rebuke by seeing him dead.
We have some idea how we got to the point that same-sex "marriage" became a possibility. We don't yet know where exactly it will lead. All we can know is that, like St. John, we must be willing to defend marriage as a sacred institution, no matter the cost. Doubts and difficulties may plague us in these trials, as they did John, but our Lord's words can comfort us as they comforted John in the face of impending martyrdom:
John, calling to him two of his disciples, sent them to the Lord, saying, "Are you he who is to come, or shall we look for another?" And when the men had come to him, they said, "John the Baptist has sent us to you, saying, 'Are you he who is to come, or shall we look for another?'" In that hour he cured many of diseases and plagues and evil spirits, and on many that were blind he bestowed sight. And he answered them, "Go and tell John what you have seen and heard: the blind receive their sight, the lame walk, lepers are cleansed, and the deaf hear, the dead are raised up, the poor have good news preached to them. And blessed is he who takes no offense at me" (Luke 7:19–23).