In the End: Saul was not saved.
In verse 13 in 2 Samuel 7:12-16, we see that the "he" who built a house in the Lord's name is talking about Solomon. In verse 15, it says that the Lord' mercy shall not depart from him [Solomon] as the Lord took away from Saul.
2 Samuel 7:12-16
12 "And when thy days be fulfilled, and thou shalt sleep with thy fathers, I will set up thy seed after thee, which shall proceed out of thy bowels, and I will establish his kingdom.
13
He shall build an house for my name, and I will stablish the throne of his kingdom for ever.
14
I will be his father, and he shall be my son. If he commit iniquity, I will chasten him with the rod of men, and with the stripes of the children of men:
15
But my mercy shall not depart away from him, as I took it from Saul, whom I put away before thee.
16
And thine house and thy kingdom shall be established for ever before thee: thy throne shall be established for ever."
In addition to that, 1 John 3:15 says that no murderer has eternal life abiding in him. Saul committed self murder by taking his own life. Saul also indulged in consulting one who contacted the dead and he did not seek forgiveness of such a sin with the Lord, either.
"And the soul that turneth after such as have familiar spirits, and after wizards, to go a whoring after them, I will even set my face against that soul, and will cut him off from among his people."
(Leviticus 20:6).
We learn that even God's Spirit had departed from Saul. Saul had desired to hear from God, but God would not answer him. This means Saul did something so bad that he could not be forgiven. Saul went down the path of sin to his own destruction. A person has to confess of their sin in order to be forgiven of their sins (1 John 1:9). A person cannot confess of their sin when they are dead. By then, it is too late. We know in the story of Lazarus and the Rich-man that the rich-man was condemned for not helping the beggar Lazarus at his own gates. No doubt he was sorrowful and wished he could have changed his decision. But it was too late.