For the Spanish at least, the big sticking issue was did the natives, as heathens, have title to their land? The issue was solved by "el Requerimiento" issued in 1513. It basically provided for the Spanish seizure of all native land on the grounds that they were not Catholic. I quote here from the edict

"Wherefore, as best we can, we ask and require you that you consider what we have said to you, and that you take the time that shall be necessary to understand and deliberate upon it, and that you acknowledge the Church as the ruler and superior of the whole world,

But if you do not do this, and maliciously make delay in it, I certify to you that, with the help of God, we shall powerfully enter into your country, and shall make war against you in all ways and manners that we can, and shall subject you to the yoke and obedience of the Church and of their highnesses [Ferdinand and Isabella]; we shall take you, and your wives, and your children, and shall make slaves of them, and as such shall sell and dispose of them as their highnesses may command"

The land was then parceled back out to Spanish colonists who would then group the natives in to Encomienderos to use as a labour force. The problems came more often when the natives had been "converted" to Christianity. Surely as pious Christians there was no ground to take their land from them? This view was most often expressed by the priests, Dominicans mostly I think, who went to the Americas and learnt the native languages and so on. The counter argument being that the natives were not very good Christians and "had not entered into full knowledge of Christ" and could therefore still be enslaved and have their land taken from them.

Empires of the New World. J.H. Elliot

Adding to the answer of /u/azdac7 , the legal aspects was in a sense determined by the simple fact that the natives were not Catholic, hence just like in the Reconquista, the Spaniards were given rights through Papal Bulls to conquer those lands. The Reconquesta model is what was largely followed in the exploration and conquest of the Americas. Conquistadors were promised the control and benefit of lands that they explore. In return they swear fealty to the crown and follow the theoretical laws and pay their taxes.

The crown itself had little control over its agents abroad, nor the ability to execute their own ventures. Thus, there was a recurring theme of administrators and religious structures being imposed after the conquistadors already established themselves years or decades ago.

As for the natives themselves there were arguments as to how they should be treated, or to be precise whether they deserved to be sold into slavery, and how they would be made to convert to Catholicism. Isabella decreed that they were to be treated with love, but she was vague on the details. Sepulveda espoused the view that they deserved natural slavery because of their backwardness, a different rationale than what was used for African slavery, which is that they had rejected Christianity, so they deserved perpetual bondage.

Many others, including Las Casas, argued that the natives had not heard the word of God, so how could they be guilty of having rejected it? Hence, The Requirement.

Hope this gives a better context as to how The Requirement came about.