While we know little about Matthias' actions after this event (and Scripture says little about his life beforehand), it isn't simply that men chose Matthias and the Spirit chose Paul. There is at least one very important consideration that should lead us away from that conclusion.
If we conclude that men chose Matthias, then we must either ignore the fact that the 120 prayed before and during the selection process or conclude that God misled them.
The 120 were praying constantly in the days between Jesus' ascension and the appointment of Mattias.
Acts 1:14 All these continued together in prayer with one mind, together with the women, along with Mary the mother of Jesus, and his brothers.
Acts 1:15 In those days Peter stood up among the believers (a gathering of about one hundred and twenty people) and said,
Peter then recounts the ministry of Judas and how he fell away. Peter concludes his speech by applying the Psalms to the current event:
Acts 1:20 “For it is written in the book of Psalms, ‘Let his house become deserted, and let there be no one to live in it,’ and ‘Let another take his position of responsibility.’
Peter has been in prayer. The 120 have been in prayer with him. The rest of the Eleven are there and have been praying. While one man might conclude on his own that they need to fill out the Twelve, it seems less likely that 119 others (who have likewise been in prayer) would go along with it unless the Lord led them that way. We see nothing of the other Twelve arguing against Peter (even though James and John, the other of the Top Three were there that day). Surely, if this were not God's will, then one of the Eleven would have spoken up. Luke, who later records that there is great dissension around Paul's selection and throughout his ministry, would surely have recorded if Peter had been in the wrong. Indeed, Luke records how Peter needed to be told one thing three times by God himself!
After the nominations, the prayer continues and the 120 put their faith in the God guiding the lot (Proverbs 16:33):
Acts 1:24 Then they prayed, “Lord, you know the hearts of all. Show us which one of these two you have chosen..."
However, after James is martyred in Acts 12:2, there is no discussion of replacing him. I conclude that on this day, Peter had heard from God that they needed a twelfth man and that God guided the selection of Matthias. I find it significant that even though the Twelve could have simply appointed another, they took the matter to the 120 for all to pray over.
While little is known of Matthias, we do know that Matthias was with them from the beginning (Acts 1:22), according to Eusebius, he was one of the seventy sent out by Christ (Church History 1.12.3). We also know that Jesus himself chose the original Twelve. In Jesus' eyes, the Twelve Apostles were to rule over Israel in the days of God's triumph (Matthew 19:28, Luke 22:30). This band of twelve represented the remnant prophesied many times in the Old Testament (Isaiah 8:16-18; 10:20-22; 11:11-16; 28:5; 37:4, 31-32; 49:6; Jeremiah 23:3-8; 31:7-23, 27-34; Micah 2:12-13; 4:7).
This appointment is presented just before the day of Pentecost. When the time came to preach first to those in Jerusalem, to show that they were the remnant, the number twelve needed to be complete. As LoveTheFaith points out, Acts 2:14 says that Peter stood up with the eleven, meaning that Matthias witnessed on that day as one of the Twelve.
The need to appoint another and Matthias' elevation came from God for a reason. Even though we are told nothing else of his ministry in the New Testament after Pentecost,1 we cannot conclude from that fact that he was an apostle appointed only by men. In fact, with the amount of prayer before, during, and after the selection, they must have heard from God.
1The New Testament likewise says nothing of the ministry of several of the Apostles chosen by Jesus himself (for example, Thaddeus and Simon the Zealot).