How do we know that Picard didn't try to simply stun his double? In most instances in which we see phasers set to kill, the target is vaporized. Perhaps the double's death had to do more with his weakened state than it did with Picard's intent.
Well... do you recall what he said right before he activated his hand phaser?
"
I cannot allow you to leave the ship. In order to move forward, the cycle must end."
End seems pretty plain to me.
I recall it vividly as it's my favorite TNG episode. But "the cycle must end" implies nothing about phaser settings or intent to kill or stun. For the cycle to end, neither Picard can leave the ship. The stunned double is just as unlikely as a dead one to leave the ship prior to the new plan being enacted.
We are going under the premise that Picard
supposedly stunned his future self. Why not on kill setting?
We do not see Picard adjust the setting on the phaser to make sure it is on stun. Therefore, either all phasers on board the Enterprise are defaulted set to stun, or he fired on the kill setting.
Since in the show, it is ordered to always 'set phasers on stun', I'm under the assumption that they are programmed by default to be on the 'kill' setting.
Here is an example of a 'stun' shot, according to the TNG. Episode: "A Matter Of Honor", Season 2. The Captain of a Klingon vessel was tricked into being beamed aboard the Enterprise, attempted to kill Federation personel, and was shot on a stun setting. Note the rings of energy, no smoke or sparks indicating energy penetration.
Now, lets look at "Time Squared", the focal point of the argument. Picard, realizing that he cannot allow his future self to leave the ship and re-loop the timeline, he takes a phaser immediately from the shuttle hangar arsenal, shows no signs of attempting to adjust or verify its settings, aims at his target 'Center Mass', and with highly emotional music in the background, fires.
Note the infliction of the bolt below the chest cavity, near midline of the interior portion of the body, on the victims (R) - right side of his Xyphoid Process, where the liver and part of the stomach would be. Also note the sparks, and smoke as the phaser bolt penetrates the body. This is NOT an indication of a non-penetration wound.
The future Picard was suffering from a diluted, emotional trauma, having seen the destruction of the Enterprise, as well as his fellow crew members killed in the process. This Picard is living in the future and present at the same time. It's like Troi explained: "
This Picard is on the other side of a canyon, seeing us only though a mist of illusion and nightmare, unable to speak to us (or something to that effect)"
This is not the same Picard we know. It is a broken man we see. Set on only one objective, not trying to find other options.
I could make just as a valid argument that Picard killed his future self out of mercy as you can make an argument that Picard meant to stun his future self, and accidentally killed him in the process (does this mean he committed manslaughter on himself? -joke-

)