I can't give a full answer on how powerful the empire was, but it was certainly not solely attributable to the discovery of the new world.

Spain was infact already a major power before discovering the Americas, and their quest for the new world is really a consequence of how well Spain was doing at the time. Prior to discovering the new world, Spain had been riding a series of highs already:

• The Spanish kingdoms had, over a period of 800 years prior to the discovery of the Americas, systematically recovered most of the Iberian peninsula from the Muslim kingdom of Andalusia. 1
• After the reconquista, the peninsula counted four kingdoms: Portugal, Castille, Aragon, and the tiny basque kingdom of Navarra. At the end of the reconquista, Aragon and Castille entered a personal union when Isabella of Castille married Ferdinand II of Aragon. Their marriage united two of the most powerful kingdoms in roughly what we would call Spain today. 2
• It was Isabella and Ferdinand who commissioned the exploration of the Atlantic in the hopes of finding a westward route to India. They funded, among many others, the expedition of Colombus. 3
• Ferdinand would, throughout his rule, also claim the crowns of Naples and Sicily in southern Italy, the kingdom of Navarra in northern Spain, and incorporate Sardinia into his realm.
• The culmination of the successful rule of Isabella and Ferdinand would later culminate in their grandson, Charles V, Holy Roman Emperor, and probably the most powerful man of his time (Suleiman the Magnificent was the only real contender in this part of the world). His power was threatening enough to his neighbours that it drove the French and Ottoman monarchs to forge the so-called "unholy alliance".

Most consider Spain's apogee to be the rule of Charles V or his successor Philip II, both ruled over an empire upon which the sun never sets. This period saw Spain be considered as the strongest land power in Europe, with significant victories against France (Francis I the king of France was even captured in the Battle of Pavia!), German Protestants, Dutch rebels (the Duke of Alba marched tens of thousands of Spanish troops from Italy to the Low Countries, a great feat even for contemporary armies!), Ottoman invasion of Austria,and even Portugal when power vacuum led Philip II to invade it and bring it under personal rule.

At sea, the middle part of Philip II's reign saw Spain's Mediterranean fleet, along with her allies, stop Ottoman advance at the Siege of Malta and eventually the great Battle of Lepanto. This was a great accomplishment that set the tone for the next century of Spanish security in the western and central Mediterranean, allowing them to keep the Balearic islands, Genoa, Naples, etc. as important holdings and allies.

Later on, Philip II was able to use his power and influence to support the Catholic party in France's Wars of Religion, which was a key concern for Spain's security. Although he did fail in invading England, and he had to declare bankruptcy several times.

More specifically, how did silver from the New World help Spain's rise? It helped secure funds and alliances, importantly from Genoa as a naval partner, from Italian and German land forces, and from German and Genoese bankers to provide means to pay its armies and navies scattered around Europe. But the silver was not the only guarantee of financial fitness. Another very important factor is Castile's well-established taxation system, as the Cruzada tax was used for Spain's imperial ventures. This tax was very stable, unlike New World silver which in the beginning took a while to come to its full potential.

So among historians, there are obviously several camps. One is led by Henry Kamen, whose thesis is that Spain never rose to be a true empire in the first place, as it is truly a composite monarchy and that a lot of the heavy lifting is done by its "allies" or "constituent subjects". "But Columbus was from Genoa, Magellan was Portuguese, many conquistadors were Italians or Dutch, and most of Spain's armies in Germany were paid German or Italian mercenaries," are lines used.

Another is led by John Elliott and Geoffrey Parker, who focus on Spain's imperial overstretch. Spain may have been the first among Europeans to discover and control parts of the New World, but their commitments increased and eventually they were left vulnerable. Parker's "The Grand Strategy of Philip II" is a great book to read, as it explains the strategic focus on Spain shifting to contend with crises as they arise, unlike many English-language pop culture that hyper-focus on the failed Spanish Armada.

So to summarize, what are some of the factors that led to Spain's rise:

• Infrastructure, logistics, and traditions from the Reconquista was successfully used elsewhere. This includes the Cruzada tax, inclusion of minor nobility as officers in the armies, military experience, and strong royal rule at the cost of local nobility.

• Financial viability. Which is a combination of new world riches and the steady and dependable income notably from Castile and the Cruzada tax.

• Partnerships. Genoa provided naval power in the Mediterranean, Portugal in the Atlantic and eventually Indian and Pacific oceans. Italian states became key providers of army manpower, as did German states.

• Weaknesses in key adversaries. France plunged into religious wars, Ottoman had to face growing threat to its east, and strengthening central/eastern European powers.

• Luck. Charles V's inheritance, Portuguese succession crises, and successes in some of the ventures, are attributable to luck.

This is a big question and I think you'll have better answers if you focus on what exactly you mean by "big and powerful": economically, or militarily, and compared to what. Even better, if there is a particular era you are interested in.

To give better context, you may like to read through several related discussions in the past:

• Why was Spain not further united?

• Was the Spanish empire doomed from the start?

• Why did Spain not keep up with other powers?