Falling upon your sword when you have been fatally wounded, knowing that your time is at hand (from God), is not an act of suicide. Everything was taken from him, his anointing, his prophet and the kingdom of Israel, including now three of his sons in the battle against the Philistines. Saul died in battle as did his three sons and his armorbearer. David never mentions anything as to Saul's death that would relate to him taking his own life.
1 Sam 31:3-5
The battle became fierce against Saul. The archers hit him, and he was severely wounded by the archers. Then Saul said to his armorbearer, “Draw your sword, and thrust me through with it, lest these uncircumcised men come and thrust me through and abuse me.” But his armorbearer would not, for he was greatly afraid. Therefore Saul took a sword and fell on it. And when his armorbearer saw that Saul was dead, he also fell on his sword, and died with him. So Saul, his three sons, his armorbearer, and all his men died together that same day.
The only thing that 'falling upon his own sword' suggests is that Saul and Israel were defeated in battle. We should not make anything more of that, especially concerning suicide. For the issues of life and death belong to God and Saul was told that he would die in battle against the Philistine the very day he did.
1 Sam 31:3-5
The battle became fierce against Saul. The archers hit him, and he was severely wounded by the archers. Then Saul said to his armorbearer, “Draw your sword, and thrust me through with it, lest these uncircumcised men come and thrust me through and abuse me.” But his armorbearer would not, for he was greatly afraid. Therefore Saul took a sword and fell on it. And when his armorbearer saw that Saul was dead, he also fell on his sword, and died with him. So Saul, his three sons, his armorbearer, and all his men died together that same day.
The only thing that 'falling upon his own sword' suggests is that Saul and Israel were defeated in battle. We should not make anything more of that, especially concerning suicide. For the issues of life and death belong to God and Saul was told that he would die in battle against the Philistine the very day he did.