1. Peter had the authority to choose another apostle
2. Matthias was an apostle
Peter chose Matthias but Jesus chose Paul.
Who had authority?
All four had authority.
Matthias was not chosen by Peter, he was chosen by casting lots....
Or do you mean anyone hallucinating Jesus during heat stroke is automatically an Apostle? (which is the secular description of what happened to Paul).
Or do you mean he was called by his vision as described in ACTS.
Suddenly a light from heaven flashed round about him. He fell on the ground and he heard a voice saying to him, `Saul, Saul, why do you persecute me?' He said, `Who, are you, sir?' He said, `I am Jesus whom you are persecuting. But rise; go into the city, and you will be told what to do.'
So he went to Christians to get healed.
From Barclay's bible study (a protestant book)
This is Luke's account of what happened to Paul after his conversion. If we want to have the chronology of the whole period in our minds we must also read Paul's own account of the matter in Gal.1:15-24. When we put the two accounts together we find that the chain of events runs like this. (i) Saul is converted on the Damascus Road. (ii) He preaches in Damascus. (iii) He goes away to Arabia (Gal.1:17). (iv) He returns and preaches in Damascus for a period of three years (Gal.1:18). (v) He goes to Jerusalem. (vi) He escapes from Jerusalem to Caesarea. (vii) He returns to the regions of Syria and Cilicia (Gal.1:21).
In other words, he preached locally but wasn't an apostle.(Ac.4:36-37). Again, from Barclay:
When Barnabas saw the Gentiles being swept into the fellowship of the Church he was glad; but he recognized that someone must be put in charge of this work. That someone must be a man with a double background...
So essentially it was Barnabas who "appointed" Paul to preach to the gentiles. I could go on, but I advise you read it for yourself.
Peter didn't choose. They prayed that God would choose.
Acts 1:24 And they prayed, and said, Thou, Lord, which knowest the hearts of all men, shew whether of these two thou hast chosen,
Once again Catholics have tried to give man credit for what God has done. It's called blasphemy.