Jesus said he was sent only for the lost sheep of Israel, so what is their destiny, according to Catholicism?

There were quite a few Jewish followers of Jesus from the very beginning. Immediately after the resurrection, the Bible records 3,000 Jews becoming believers in a single day. 1 and 2 Peter were both written primarily to Jewish believers as was "Hebrews", obviously. So, there were a good number of Jews who did believe in Jesus at the outset.

It is true, though, that may rejected Jesus as their Messiah, and this has been true throughout history. Both Jews and Gentiles reject Jesus. However, in the last 50 years, it is true that the number of Jewish believers has exploded. There are an estimated 150 Messianic Jewish congregations in Israel today with 20,000 Jewish believers. This is four times as many as there were in 1997. Source

The question, however, really centers on what happens to those who reject Jesus. The answer to this does not differ based on whether one is a Jew or Gentile, though.

The apostle John clearly gives the answer in this verse:

Whoever denies the Son does not have the Father; the one who confesses the Son has the Father also. 1 John 2:23 NASB

So, whoever denies the Son does not even have a relationship with God at all. Jesus affirmed this as well:

For not even the Father judges anyone, but He has given all judgment to the Son, 23 so that all will honor the Son even as they honor the Father. He who does not honor the Son does not honor the Father who sent Him. John 5:22-23 NASB

So, according to Jesus Himself, if one does not honor the Son, it is impossible to honor the Father.

Ultimately, then, a person's eternal destination depends on whether to accept or reject Jesus:

“For God so loved the world, that He gave His only begotten Son, that whoever believes in Him shall not perish, but have eternal life. 17 For God did not send the Son into the world to judge the world, but that the world might be saved through Him. 18 He who believes in Him is not judged; he who does not believe has been judged already, because he has not believed in the name of the only begotten Son of God. John 3:16-18 NASB

So, yes, anyone who rejects God's promised Messiah, God's Passover Lamb, God's sacrifice for sin, God's offer of forgiveness--that person is destined to pay for his own sin. God is a just God and does require a payment for sin. He offers to pay it on our behalf, but He does not force us to accept it.

"Israel" is more than the ethnic Jewish race, according to the New Covenant that was initiated by the death and resurrection of Jesus. The name originates from the time the patriarch Jacob wrestled with 'the' Angel and was renamed 'Prince of God'. It became the identity of the nation that descended from the sons of Jacob, the 'twelve tribes of Israel'.

John the Baptist asserted that God could raise up sons of Abraham from the stones, meaning that an Israelite could be defined by something other than his genetic code (Matthew 3:9; Luke 3:8).

The 'House of Israel' Jesus came to save referred to anyone who comes to God in faith, who believe in God's (not man's) plan of redemption through Jesus. Paul goes on to state that ethnic Jews who rejected God's Son were removed from the tree of Israel, and their places were being taken by non-Jews who entered into the house of Israel by faith in Him (Romans 10).

The 'lost sheep of Israel' refers to anyone, Jew or Gentile, who recognized their separation from God, their powerlessness to change it through their own efforts (the law), and accepted God's gracious answer in Jesus. "But to all who believed him and accepted him, he gave the right to become children of God." (John 1:12 NLT)