This is a question of opinion at the end of the day, because people will interpret the history differently. The answer will also depend on whether you're looking at the intentions behind the Crusade (along with the various sub-categories that go with that - the Pope's intentions, the lords' intentions, the commoners' intentions, etc) or the reality of the Crusade. In any event, the short answer is no, I would not describe the First Crusade as a pilgrimage with a touch of war. It has been described as an armed pilgrimage before, but I feel like saying it just had "a touch of war" discounts the tremendous bloodshed and the centuries of unrest that followed.

The first and foremost reason is that virtually nobody at the time saw it that way. Consider this excerpt from Pope Urban II's famous sermon in which he first preached the Crusade:

Although, O sons of God, you have promised more firmly than ever to keep the peace among yourselves and to preserve the rights of the church, there remains still an important work for you to do. Freshly quickened by the divine correction, you must apply the strength of your righteousness to another matter which concerns you as well as God. For your brethren who live in the east are in urgent need of your help, and you must hasten to give them the aid which has often been promised them. For, as the most of you have heard, the Turks and Arabs have attacked them and have conquered the territory of Romania [the Greek empire] as far west as the shore of the Mediterranean and the Hellespont, which is called the Arm of St. George. They have occupied more and more of the lands of those Christians, and have overcome them in seven battles. They have killed and captured many, and have destroyed the churches and devastated the empire. If you permit them to continue thus for awhile with impurity, the faithful of God will be much more widely attacked by them. On this account I, or rather the Lord, beseech you as Christ's heralds to publish this everywhere and to persuade all people of whatever rank, foot-soldiers and knights, poor and rich, to carry aid promptly to those Christians and to destroy that vile race from the lands of our friends. I say this to those who are present, it meant also for those who are absent. Moreover, Christ commands it. ... With what reproaches will the Lord overwhelm us if you do not aid those who, with us, profess the Christian religion! Let those who have been accustomed unjustly to wage private warfare against the faithful now go against the infidels and end with victory this war which should have been begun long ago.

The TL;DR of that is basically, "Hey, Christians, some Muslims have conquered Christian territory in Greece and Christ commands you to go take it back." The territorial changeover he's referring to is the conquest of some fairly significant chunks of the Byzantine empire in Anatolia, held by Kilij Arslan on behalf of the Seljuk Turks. The original intention of the Crusades, as eluded to in that chunk of sermon by Urban, was to go and assist the Byzantine emperor, Alexius Komnenus, in reclaiming that lost territory.

Whether Urban was really intending with this escapade is a subject of some debate, but ultimately his true intentions are irrelevant. You can see that, as it was originally preached, not only was the First Crusade not a pilgrimage, it wasn't even supposed to involve the Holy Land. The lords who served as the generals of the Crusade don't appear to have been motivated by piety either. As soon as he laid eyes on the group of leaders (including one whom he had literally just fought, and lost, a territorial war with in Sicily), the first thing he did was make them sign what was essentially a contract promising that all lands captured would be ceded back to the Byzantine empire - an agreement honoured by exactly nobody.

Once the Crusaders reached the Holy Land, already outside the scope of their initial purpose, they fought several battles including a number sieges at Nicaea, Antioch, Ma'rra, Dorylaeum, and of course Jerusalem among many others. They arrived in Asia Minor in 1097, by the time they captured Antioch in 1098, Stephen of Blois wrote to his wife that they had captured 165 cities and fortresses - more than a touch of warfare, I think it's safe to say. Jerusalem fell to the Crusaders in 1099 and the Latin States, headed up by the Kingdom of Jerusalem, were established shortly thereafter. Here is a map of the territory that comprised the Latin States. It's not a small chunk of land as you can see.

The size of the territory is indicative of the Crusaders' hopes for it. They were well beyond the bounds of a pilgrimage and getting into full-blown invasion and resettlement. That part of the equation never worked out quite as well as they'd hoped - the Europeans were always a minority in their own territory and struggled to entice significant numbers of their kinsmen and countrymen to come to the Holy Land and stay for good. Nevertheless, they had their hopes and intentions, and those were for a permanent European dominion in the Middle East.

I don't mean by this post to suggest that there was no pilgrimage aspect involved in the Crusades. There were definitely elements of it at play, including the pursuit of Jerusalem as a final destination and the frantic search for relics along the way, which produced artifacts that were claimed to be the cross Jesus was crucified on, the spear that pierced his side, and so on and so forth. The religious side of it was certainly present and it would be foolish to discount the role it played in the Crusades. However, I think it's equally foolish to discount the role that ambition, politics, and economy played in the First Crusade's development as well. We're talking about an endeavour that spanned years, cost countless lives, and dramatically altered the balance of power in the Muslim world for centuries. There was a pilgrimage within it, but to ignore the war for the pilgrimage feels to me like a shallow and unfair reading of a tremendously complicated event.

Sources: The primary sources I referenced can both be found in the Online Medieval Source book. Respectively, they were:

Urban II: Speech at Council of Clermont, 1095, according to Fulcher of Chartres, by Fulcher of Chartres

Stephen, Count of Blois and Chartres, To hiis Wife Adele by Stephen of Blois

Other information comes from "The Crusades Through Arab Eyes" by Amin Maalouf and "The Crusades" By Hans Eberhard Meyer.