Gasparovic, 63, Meciar's former right-hand man, scored a decisive victory in Saturday's presidential election, taking 59.9% of the vote compared to Meciar's 40.1%.
His surprise victory came after an election which saw the original favourite, foreign minister Eduard Kukan, knocked out in the first round on 3 April.
Meciar, the controversial former prime minister who negotiated the split of Czechoslovakia in 1993, had been widely expected to win the runoff due to his hardcore of faithful followers.
But the centre-left Gasparovic over the past two weeks rallied support from opponents of Meciar, arguing he was more acceptable to foreign partners.
"I am accepted abroad," Gasparovic told Slovaks before voting began.
Slovakia joined NATO on 2 April and is among 10 countries due to join the European Union on 1 May.
Gasparovic served Meciar and his People's Party-Movement for a Democratic Slovakia party for a decade before breaking off to found his own party after they fell out in 2002.
He was supported in his presidential attempt by opposition party Smer, the most popular political party in the country.