Can you show us the passage context that makes you believe that it is up to anyone other than God?
It's God who saves so unless He OK's it no one gets saved. The question is does He choose individuals to be saved. He has chosen to save those who will believe. So, in that sense God has chosen. However, He hasn't chosen who will be believe. There are quite a few passages that speak to the issue. One comes from the apostle John speaking of why Jesus came.
6 There was a man sent from God, whose name was John.
7 The same came for a witness, to bear witness of the Light,
that all men through him might believe.
8 He was not that Light, but was sent to bear witness of that Light.
9 That was the true Light, which lighteth
every man that cometh into the world.
(Jn. 1:6-9 KJV)
In this passage John says that John the Baptist came as a witness to Jesus who was the light. In the Bible light is often a metaphor for understanding or illumination. John writes that all men through Him might believe. Then in verse 9 he gets specific. He says that Christ gives light or understanding to every man who comes into the world. He uses the singular for man. It means each individual man is given some understanding so that he might be saved. In verse 7 where John says, that all through Him might believe, he puts the Greek word Pisteuo, in the subjunctive mood and the active voice. The subjunctive mood indicates possibility or probability, but not certainty. The active voice indicates that the action of believing is being done by the subject, which in this case is all men. So, he's saying that all men have some understanding and that they might be saved. He's also saying that the believing is done by them not to them.
Another passage is from Paul's words. In Acts Paul tells us that certain Jews rejected salvation.
43 Now when the congregation was broken up, many of the Jews and religious proselytes followed Paul and Barnabas: who, speaking to them, persuaded them to continue in the grace of God.
44 And the next sabbath day came almost the whole city together to hear the word of God.
45 But when the Jews saw the multitudes, they were filled with envy, and spake against those things which were spoken by Paul, contradicting and blaspheming.
46 Then Paul and Barnabas waxed bold, and said, It was necessary that the word of God should first have been spoken to you: but seeing ye put it from you, and judge yourselves unworthy of everlasting life, lo, we turn to the Gentiles.
47 For so hath the Lord commanded us, saying, I have set thee to be a light of the Gentiles, that thou shouldest be for salvation unto the ends of the earth.
48 And when the Gentiles heard this, they were glad, and glorified the word of the Lord: and as many as were ordained to eternal life believed. (Acts 13:43-48 KJV)
Here Paul plainly states that these Jews rejected the word of God and they judged themselves unworthy of everlasting life.
