The answer to this question is not only yes, but he has already forgiven the world of all sins, past, present and future. There is only one unforgiveable sin and that is rejecting Christ, or blasphemy. Blasphemy of the Holy Spirit is rejecting the truth of Christ, denying it even though you have a conviction of the truth of Christ work on the cross. You are literally saying, I know the truth, but reject it. John 16: 9 tells us that the Holy Spirit will convict of Sin, but many will not believe in Christ. Jesus death paid for every sin except the rejection of his work on the cross. If you do not believe that God, through Christ reconciled the whole world through Christ due to them being in Adam, and ignore the Holy Spirits prompting of that truth, then you will die and go to Hell. The sad part of the whole thing is, God forgave that person already and they refused to accept or believe it!
The key here is will you accept it. Many Christians believe that when they receive Christ that the sins they have committed up to that point are forgiven, but then the sins they commit after they are saved are not and have to continually go to God and be re-cleansed and confessed for God to love and accept them. That is the farthest from the truth. If you read II corth 5: 17-21 and Romans chapter five and I john 1: 9 you find a startling fact. God loved you and I while we were still sinners. His forgiveness of sins and his wrath for sin was laid on Jesus Christ not only you, but for all men. The problem, is, many men will not come to Christ and accept that sacrifice. God has reconciled the whole world to himself through his son if they will receive it. As a Christian you will commit sins and sin, but God has already forgiven you of that sin from the foundation of the world. I John 1: 9 is written for Christians, so they can get the condemnation sin brings off them. When it says If you confess your sins, God is faithful and Just to forgive you of your sin and cleanse you from all unrighteousness. It does not mean you are made unrighteous when you sin, it means when you go to God and confess that sin, you are acknowledging the finished work of Christ on the cross. When you sin, you should approach God with this prayer using I john 1: 9 as a guide. You say, father, I have sinned and I confess that sin before you. I thank you that Jesus bore the penalty for that sin on the cross for me and I yield my self to the cleansing blood of Christ which continually washes me and keeps me free from sin. What happens when you do that is your conscience which is defiled by your unrighteous acts of sin is then cleansed so that your self condemnation does not keep you separated from God.
The first thing you do when you sin is feel a separation from God. Its not God separating from you, its your defiled conscience making you feel that way. Nothing can ever separate you from the love of God and he has told us he will never leave or forsake us. Rom 6: 1 tells us that if we walk according to the spirit we will not walk in condemnation because there is no condemnation for those who are in Christ Jesus. If you go on and read I John 3: 18-23 it tells us that when our heart condemns us we have no confidence before God. The reason you run to God when you sin is to apply his promises to heal your condemned heart. Vs 21 tells us that if our hearts do not condemn us, then we have confidence toward God. When we sin, we feel condemned and have no confidence before God. If you go on to I john 5: 14-15 it states that when we have confidence before God then we can easily ask for anything and receive it because our hearts are not condemned. Adam and Eve really show us this. When they sinned they ran from God, yet he was searching for them. He was not angry at them, but they figured he was because they had disobeyed him. If you will read, when Cain killed Able, God was right their talking to Cain. No, sin caused men to drift further and further away from God even though God kept drawing close to them. It was men who pushed God away, not God who left us. As men go their own way without God, it causes the seperation. God never wanted to be seperated from men, it was our foolish decisions that caused the seperation. When you get to Noah, you really see this. Men had gone so far from God that they could not hear from him. There was only eight people who heard from God which goes to show you that when you have a relationship with God, you can still sense his closeness in a fallen world!
This is why it is so important to understand the New Testament of Grace. The law condemns and strengthens sin, but Grace through faith brings restoration even when you blow it.
Now, sin has a consequence and it brings death into your lives. Stay away from sin; it will condemn you to the point of you running from God. Its wages will destroy you.