[Tar Elenion]

Elrond was in the War of Wrath:
"Thereupon Elrond paused a while and sighed. ‘I remember well the splendour of their banners,’ he said. ‘It recalled to me the glory of the Elder Days and the hosts of Beleriand, so many great princes and captains were assembled. And yet not so many, nor so fair, as when Thangorodrim was broken, and the Elves deemed that evil was ended for ever, and it was not so.’"
LotR, Council of Elrond
Led armies in the War of Elves and Sauron:
"When news of this reached Gil-galad he sent out a force under Elrond Half-elven; but Elrond had far to go, and Sauron turned north and made at once for Eregion."
"But his force was weakened by the necessity of leaving a strong detachment to contain Elrond and prevent him coming down upon his rear."
"The army that was besieging Imladris was caught between Elrond and Gilgalad, and utterly destroyed."
UT, History of G&C
(along with the participation in the Last Alliance)
This seems to establish Elrond as a warrior and war-leader in the First and Second Ages.

In Third Age draft materials Tolkien also indicated Elrond took part in the Angmar wars, but replaced him with Glorfindel.

In my opinion, Elrond as a healer is being established in the Third Age. Hence abstaining from war, as do elven-healers (though male healers will go to war at last need).

Elf-women abstain from war and this abstention gives them great virtue in healing (though elven-women will fight in desperate defence, much as Galadriel did at the First Kinslaying).