This is a difficult question to answer, but I am going to try my best. One of the reasons for it being hard to answer is that the entire system of how law, governing, and authority were viewed in Iberia and the Americas was changing at this time. Diego would have had multiple lawyers and jurists giving him contradictory advice and counsel as some areas of the Caribbean were falling under different royal-mandated authority throughout the first decade of the 16th century.

To recap, Cristopher Columbus was granted several things under the agreement between he and the Catholic Kings at Santa Fe in the Spring of 1492. He would be given the title Admiral of the Ocean Sea, the power of Viceroy, the power of Governor, 10% of all the wealth he discovered, and was able to reinvest his profits into trade networks that gave him tax exemptions from standard Aragonese and Castilian tax laws. While the discoveries were celebrated, Columbus was not drowning in the riches he claimed would appear if one sailed west from Europe. The Queen and Columbus had a different relationship than did the Admiral and the King and it is possible that she pressured her counterpart to allow Columbus more leeway in searching for gold and access to the Asian mainland. Las Casas has famously written on the violence and exploitation of natives committed by the Admiral, but other chroniclers like Peter Martyr and Oviedo confirmed that European atrocities against natives of the Caribbean were significant. The entire family was to blame here, as Columbus's brothers and his son, Diego, were granted authority by Christopher. Settlers to Hispaniola rejected the authority of the Columbus clan, especially as it became clear that Columbus disregarded basic law and order. After the Crown was forced to pull Columbus out of the Americas, his titles were removed save for Admiral of the Ocean Sea.

So does Diego have the right to challenge royal power for taking away his father's royal concessions? The answer is in the Siete Partidas, or Seven Parts. Under medieval Spanish law (13th century, I believe) the Castilian Crown organized a body of laws that clearly defined the rights of subjects, the power and limits of the clergy, the power and limits (with sections of correct behavior and eating habits) of the monarch, basic criminal and civil law, etc. It gets complicated. One element that it holds to is that the monarch is a representative of God on earth, but must have the consent of the community to govern. By this logic, subjects can appeal royal decisions or orders through the proper channels as laid out in the Siete Partidas. At its heart, the Columbus Lawsuit merely requested that the Crown live up to its obligations of the Santa Fe agreements. This was a problem for Diego as it was clear by the first few years of the century that the Americas were not Asia. Since the Santa Fe agreements were based on Columbus reaching the Indies (east Asia) he had to find a way to argue that the elements of discovery were important and that the Columbus family be given the money that was promised from discovery. One important claim was that Columbus had mapped the coast of Central America, Veragua. Diego wanted control of that area in addition to the islands discovered in the Caribbean. All he had to do was file a grievance with the royal courts, present witnesses and documents supporting his claims, and await royal lawyers to respond with their own questions. Once the Council of the Indies was established this was much more streamlined, but in this early period it must have been incredibly time consuming.

What Diego never understood it seems, is that the Crown had been removing the power of strong individuals and families from the New World and consolidating power through royal officials who were easily replaced. Diego and his family felt that the Crown owed them for what his father had done for the Crown. When Diego married the King's cousin it forced Fernando to give Diego some concessions. He was appointed as the new governor of Hispaniola, but his power was limited. Unlike his father and uncles, Diego was forced to operate within a newly developed Caribbean structure that put clerics, judges, and royal officials in positions that the Columbus family had enjoyed during the early years of the European advance into the New World. During his tenure, the courts in Spain ruled that Diego should hold the title and financial concessions promised his father, but that the Crown could establish legal barriers to prevent Diego from holding more political power. Diego, in effect, had basic political control over several islands, but he was not as powerful as he might have wished. The Crown removed Diego from office in 1515 and again in 1523 and he died in 1526. His wife seems to have worked hard with her cousins Fernando and his grandson Carlos I to find an acceptable solution to the family problem. In 1536 the courts allowed the Columbus family to pass on the title of Admiral, gave them the title of Duke of Veragua in perpetuity, some land in Jamaica, and 8500 pesos of gold a year in compensation.

So why did Carlos I concede to the Columbus family? First, he didnt. The 1536 ruling was done under arbitration. Second, the political authority of the New World was flowing through the Council of the Indies by this time and not through governors or viceroys. Columbus's dictatorial powers in the early years of discovery could not happen again. Third, (and this is a personal theory of mine) the political climate of the Americas was getting close to restoring Columbus-like powers to certain conquistadors. Certainly Cortes had wielded more power than he should have and Peru was about to break out into civil war over the Pizarro family's hold on political and economic control of the province. By finding a resolution with the Columbus family, Carlos I sent a message to other contract holders with the Crown in the New World. Spain would do what it could to honor all contracts, but there had to be limits. Those limits would be decided by the Crown or by royal officials and their word was final.

Sauer, The Early Spanish Main

Edwards, The Spain of the Catholic Monarchs

MacLachlan, Spain's Empire in the New World

Nichols, "Las Siete Partidas" in California Law Review (A bit dated with its 1932 publication, but a concise description of the development and basic components of the law structure.