that he was shot by his own troops is quite possible, he was at the cutting edge of the PARA battleline and susequently an awful lot of british 7.62 would of been whizzing over his shoulders in an attempt to provide suppresive fire. he was probaly shot by the Argentines, as while both sides were throwing masses of 7.62 at each other with him in the middle, the Argentines were trying to hit him, while the British weren't.
that he was shot deliberately by his own troops is pretty unlikely, he was with his 'inner circle', all of whom were deeply personally loyal to him and his way of soldiering, and being relatively far from the rest of the battalion its unlikely that anyone other than his 'inner circle' would of been able to identify a particular figure as a particular individual.
working out who actually did shoot him would be impossible - assuming, as pretty much everyone does - that he was hit by 7.62X51. both sides used it as their main round, probably on a ratio of 500 7.62X51 for every one of 5.56 or 9mm.
9mm wouldn't be a candidate, its effective range is far too low to have been used in that kind of engagement. its a self-defence round/weapon, used primarily by those like support arms, helicopter crews and senior officers (who shouldn't be at the leading edge of the battle).
5.56 was used by SF, but SF - of either side - would have no interest whatsoever in being in the middle of a full-scale infantry battle.