The legislation aims to correct what the Spanish government calls the "historic mistake" of the country's monarchs Isabella and Ferdinand who in 1492 ordered Jews to convert to Catholicism or leave within the space of just weeks, under pain of death.
"Dear Sephardim, thank you for your loyalty," the king told representatives of Sephardic Jews from different countries, at the royal palace.
"Thank you for having kept like a precious treasure your language and your customs that are ours too. Thank you too for making love prevail over rancour and for teaching your children to love this country.
"How we have missed you."
There are an estimated 3.5 million Jews with Spanish ancestry around the world -- known as Sephardic Jews.
The law, which came into force last month, grants them dual citizenship.
Applicants do not have to be practising Jews but they must have their ancestry vetted by Jewish authorities and prove a "special connection" to and knowledge of Spain.
Under a previous 1924 law the government had discretionary powers to award Sephardic Jews nationality but candidates had to give up their previous citizenship and they had to be residents of Spain.
Though estimates vary, historians believe at least 200,000 Jews lived in Spain before Isabella and Ferdinand ordered them to convert or leave as part of a policy to unite the country under the banner of Catholicism.
Many found refuge in the Ottoman Empire, the Balkans, North Africa and Latin America, and those who refused to convert or leave were burned at the stake.
Justice Minister Rafael Catala said on Monday his administration had already received close to 600 demands for Spanish nationality.
In October, the government had already sped up applications for 4,300 Sephardic Jews who had filed applications for nationality before the latest law came into force.