This practice is anecdotal and circumstancial. Bear in mind that Bernal Díaz mentions the use of fat from the bodies of Indians in a context of lack of resources. He even says as much at the end of chapter LXII:

dormimos cabe un arroyo, y con el unto de un indio gordo que allí matamos, que se abrió, se curaron los heridos; que aceite no lo había; y tuvimos muy bien de cenar de unos perrillos que ellos crían

Translation: We slept by a stream, and with the lard of fat Indian that we had killed there, who had been opened, were the wounded healed; for there was no oil; and we had as a very good dinner some little dogs they raise

The mention of the lack of oil is extremely important in this context, as it means that the ordinary material for the healing of the wounds would be hot oil. In fact, in another passage Bernal Díaz says that they "burned" the wounds of the horses with the lard of an Indian that had been killed. In a different passage, Bernal says that the wounds they had suffered had been healed with oil: we came back to our camp well wounded, where we were healed with oil and tightened our wounds with linens

The burning or cauterisation of wounds with different types of fat is something well acredited in the period, and more often than not they would have had to make do. Bartolomé de las Casas, in chapter 29 of his Apologética Historia tells about the use of tortoise lard for burning wounds. I'll translate:

The priest remained very saddened for not having, unlike many others, been healed of his wounds with certain tortoise lard, burning their wounds, and in eight days many of them were healed, and those who were not healed were nearly healthy.

This was not the only possiblity for burning or cauterising wounds. The same author provides an example of wounds being burnt with oil for lack of trementine (I will abstain from commenting on how much it has got to hurt):

A surgeon started to treat him, who, for lack of the necessary trementine, burned his wounds with oil, and they were many more than had been said, and the surgeon swore that each day he found new ones

There are, of course, other classics like alcohol. It disinfects wounds, and when pure alcohol is lacking, you can always use whisky, cognac, or some other distilled spirits. This was also commonly used in the16th century, as told, for example, by Fadrique de Zúñiga in his "Libro de la cetrería":

It is considered a most certain thing that birds heal very fast from their wounds, both big and small, without other treatments besides washing their wounds with eau-de-vie, very thin, distilled three or four times through a still

Wounds can also be treated with different types of potions made from various kinds of plants, and in this respect good old Gonzalo Fernández de Oviedo comments in his Historia natural y general de las Indias on the use of "balsam" made from the "balsam tree", and says that it can heal wounds very fast, even when they are open and lacking flesh:

Going back, then, to the matter, I say that there are already many men in this island [Hispaniola] who can make this balsam, which, as many people claim, is made from small pieces of said trees, and that boiled in water comes out of them a liquor like oil or denser with the colour of clear molasses; and they use it for fresh wounds, stab wounds, or lance wounds, or any other recent wounds, for it immediately thickens the blood, and there is nothing known to be more medicinal to close wounds.