There are 5 Greek words used in the NT that are generally rendered as "gentile" They are
ἐθνικός - ethnikos - G1482
1) adapted to the genius or customs of a people, peculiar to a people, national
2) suited to the manners or language of foreigners, strange, foreign
3) in the NT savouring of the nature of pagans, alien to the worship of the true God, heathenish
3a) of the pagan, the Gentile
Part of Speech: noun masculine
A Related Word by Thayer’s/Strong’s Number: from G1484
ἐθνικῶς - ethnikōs - G1483
1) like the Gentiles
Part of Speech: adverb
A Related Word by Thayer’s/Strong’s Number: from G1482
ἔθνος - ethnos - G1484
1) a multitude (whether of men or of beasts) associated or living together
1a) a company, troop, swarm
2) a multitude of individuals of the same nature or genus
2a) the human family
3) a tribe, nation, people group
4) in the OT, foreign nations not worshipping the true God, pagans, Gentiles
5) Paul uses the term for Gentile Christians
Part of Speech: noun neuter
A Related Word by Thayer’s/Strong’s Number: probably from G1486
Ἑλλην - Hellēn - G1672
1) a Greek either by nationality, whether a native of the main land or of the Greek islands or colonies
2) in a wider sense the name embraces all nations not Jews that made the language, customs, and learning of the Greeks their own; the primary reference is to a difference of religion and worship
Part of Speech: noun masculine
A Related Word by Thayer’s/Strong’s Number: from G1671
Ἑλληνίς - Hellēnis - G1674
1) a Greek women
2) a Gentile women, not a Jewess
Part of Speech: noun feminine
A Related Word by Thayer’s/Strong’s Number: feminine of G1672
Now that we can see based on simply the definitions that the term gentile is most often referring to a nation, group of people, or even specifically the Greeks. Let's look at some scripture to set the stage before continuing.
Acts 9:15-16 NASB
But the Lord said to him, "Go, for he is a chosen instrument of Mine, to bear My name before the Gentiles and kings and the sons of Israel; (16) for I will show him how much he must suffer for My name's sake."
Acts 22:21 NASB
"And He said to me, 'Go! For I will send you far away to the Gentiles.'"
Galatians 1:15-17 NASB
But when God, who had set me apart even from my mother's womb and called me through His grace, was pleased (16) to reveal His Son in me so that I might preach Him among the Gentiles, I did not immediately consult with flesh and blood, (17) nor did I go up to Jerusalem to those who were apostles before me; but I went away to Arabia, and returned once more to Damascus.
1 Timothy 2:7 NASB
For this I was appointed a preacher and an apostle (I am telling the truth, I am not lying) as a teacher of the Gentiles in faith and truth.
Below is a clip from a teaching that Jim Staley posted. Listen to what he has to say on this subject as well.
Additionally, since Sha'ul was spreading the message of Yeshua it should also be considered what Yeshua himself said.
Matthew 10:5-6 NASB
These twelve Jesus sent out after instructing them: "Do not go in the way of the Gentiles, and do not enter any city of the Samaritans; (6) but rather go to the lost sheep of the house of Israel.
Matthew 15:24 NASB
But He answered and said, "I was sent only to the lost sheep of the house of Israel."
So, while Sha'ul would teach any and all comers, to state that his primary objective was to reach the pagan people of the known world at the time might be a mistake. Sha'ul certainly taught pagans but he was arguably focused on reaching the lost tribes of Israel to indicate that the awaited Messiah had indeed come. The Messiah that had been foretold by the prophets and that was being looked for. Sha'ul's ministry was an extension of Yeshua's ministry with a primary focus being to reach the "lost sheep of the house of Israel" but also welcoming anyone demonstrating faith in the Messiah.