King Felipe VI has called for "a new Spain that we will build together" after being proclaimed head of state in a ceremony in parliament.
Earlier, King Felipe received the royal sash from his father, Juan Carlos, at the Zarzuela Palace near Madrid.
He acceded to the throne at the stroke of midnight after King Juan Carlos formally abdicated on Wednesday.
The proceedings have been kept low key, as many Spaniards are suffering economic hardship.
The swearing-in ceremony took the form of a proclamation rather than a coronation. It is the first royal transition in Spain since democracy was restored in the 1970s.
The new king, 46, swore an oath promising to uphold the constitution.
The speaker of the lower house of parliament, Jesus Posada, then proclaimed him king, declaring: "Long live Spain! Long live the king!"
In a speech to parliament, Felipe said he had "great hope" for the future of Spain and called for unity.
"You will find in me a loyal head of state who is ready listen and understand, warn and advise as well as to defend the public interest at all times," he said.
"The monarch wants to be close to citizens… ensuring it can preserve its prestige and dignity.
"Now more than ever, citizens of Spain are rightly demanding fundamental ethical principles should govern our public life. The king should not only be a reference but who serves all citizens of Spain."
The Congress of Deputies, the lower house, stood to applaud the new king. Reports say Catalan leader Artur Mas, and Inigo Urkullu, President of the Basque government, were present but did not clap.
Mr Mas has vowed to hold a vote on independence in November while the Spanish central government has said it will block the vote.
Earlier this month, thousands of people held a demonstration in the Basque country calling for self-rule.
No foreign leaders or royal families were invited to Thursday's ceremonies.
After the proclamation, Felipe, his wife Queen Letizia and their daughters Princess Leonor and Princess Sofia were driven through Madrid's streets and the family appeared on the front balcony of the Royal Palace.
Correspondents say the new king faces a series of tough challenges if he is to restore the reputation of the monarchy.
Although Juan Carlos won plaudits for his role in restoring democracy, his image suffered when he went on a luxurious African elephant-hunting safari in the midst of the financial crisis.
His reputation suffered further damage because of tax-fraud allegations made against his daughter, Cristina, who is reported not to have been invited to the succession party.
At the same time, many Spaniards are demanding a referendum on whether to have a monarchy at all.
A demonstration is scheduled to take place in central Madrid on Thursday, the same day as the enthronement, despite a ban imposed by authorities.
The BBC's Chris Morris in Madrid says King Felipe will face the dual challenge of trying to rehabilitate the monarchy and trying to unify a country in which a vocal minority favour republicanism.
The succession has been endorsed by both of Spain's main political parties.