Spain's new King Felipe VI delivers his speech after the swearing-in ceremony. Photograph: Paco Campos/EPA

As King Felipe VI and Queen Letizia stepped out on the balcony of Madrid's royal palace, they were greeted by roaring cheers from below. The pair waved warmly; he, a multilingual royal who has spent his whole life preparing for this moment and she, a former television newsreader who local media delight in calling Spain's first middle-class queen.

"A renewed monarchy for new times," was how Felipe put it during his inaugural address as king on Thursday.

On the first day of his reign, the 46-year-old stuck tightly to a script that walked a fine line between upholding the continuity of the monarchy and suggesting subtle changes that lie ahead for Spain's embattled royal family.

The day's events were billed as low-key. The 18th-century Spanish crown and 17th-century sceptre were displayed rather than put on the king's head or in his hand, no foreign dignitaries or royals were invited and the afternoon reception for 2,000 guests featured finger foods rather than an elaborate banquet.

King Felipe VI and Queen Letizia of Spain wave to the crowds during the royal procession down the Gran Via in Madrid. Photograph: Gonzalo Arroyo Moreno/CP/Getty Images

But not all was austerity; the new king and queen were driven through the wide streets of Madrid in a Rolls-Royce with the top down, waving to the many Spaniards who spent hours waiting in the sun to catch a glimpse of them. The route was draped with Spanish flags, providing a colourful red and yellow backdrop.

In his address, Felipe promised to be a king who would "listen, understand, warn and advise".

With the country's unemployment rate persistently hovering around 25%, Felipe offered his solidarity to "those who have been hit hard by the economic crisis," and noted "the solution of their problems, particularly that of finding work, must be a priority for society and for the state".

He shied away from naming Catalonia or the Basque Country in his speech but did address the push for independence in these regions. "I want to reaffirm, as king, my faith in the unity of Spain."

Spain's King Felipe VI and Spain's Queen Letizia wave from the balcony of the Royal Palace. Photograph: Gerard Julien/AFP/Getty Images

As cameras panned over Catalan leader Artur Mas, who has vowed to hold a referendum on independence in early November, Felipe added: "In a unified and diverse Spain, based on the equality of and solidarity between its people, there is room for all of us. And for all of our feelings and sensitivities and our distinct ways of being Spaniards."

Felipe thanked his father, Juan Carlos, for his "exceptional" 39-year reign that ended on Wednesday. Once one of the world's most popular monarchs, Juan Carlos was plagued in recent years by a series of scandals that sent his approval ratings plummeting.

The 76-year-old abdicated king was absent from the ceremony, as was Felipe's sister Cristina, who, along with her husband, is at the centre of a long-running corruption investigation.

Felipe sought on Thursday to disentangle the monarchy from controversy. "The crown must constantly earn citizens' appreciation, respect and trust," he said. "Today, more than ever, citizens are rightly demanding that public officials, guided by morals and ethical principles, lead by example."

Wearing a Spanish flag around his neck, Ivan Hernandez Lillo welcomed the idea of a new, younger king. The 19-year-old saw the monarchy as crucial to the country's stability, he said, acknowledging that many of his peers disagreed. "But even the fact that they can go out and protest against the monarchy is in part thanks to the monarchy."

The new king will be a great global ambassador for Spain, said Silvia Buitrago Valbuena. "We couldn't have better representation," she said, smiling. "Much better than our politicians."

For many the day's events were a welcome distraction from Spain's other news of the day; the defending champions crashing out of the World Cup after losing to Chile on Wednesday evening.

"After what happened to our football team yesterday, this gives me a reason to be happy," said Rafael Redondo. "Our country is so diverse – from Andalusia to the Basque Country. The monarchy is the glue that holds us together," he said. Behind him a group cheered: "España, España."

After recent weeks saw tens of thousands of Spaniards take to the streets to demand a referendum on the monarchy, security was high across the capital with 7,000 police, 120 snipers and helicopters monitoring the scene.

A rally against the royals to coincide with the coronation was denied authorisation for what officials called "reasons of security and prudence". An estimated 400 people gathered for a peaceful protest anyway, but did so about half a mile away from the royal events.

In the area where the the coronation celebrations were taking place, police enforced a ban on any republican flags or symbols in an attempt to limit potential confrontations between monarchists and republicans. Three people were arrested for defying the ban, Spanish media reported.