Solyom was speaking to reporters half-way across a bridge over the Danube that links Hungary and Slovakia.
He had been due to attend the unveiling of a statue of Stephen, a medieval king of Hungary, in the town of Komarno, just over the border.
Robert Fico, the Slovak prime minister, said earlier on Friday: "Komarno lies in Slovakia's territory, it's not a Hungarian town," adding that Solyom would be welcome at any other time, but that coming on August 21 - a historically significant day - was "mere provocation".
"He wants to spark a conflict, we want to avoid it," Fico said.
Slovakia remembers the 1968 invasion by Soviet-led troops - among them Hungarians - on August 21 into then-Czechoslovakia, which ended the reform drive by the communist leadership, known as "Prague Spring".
|Solyom was speaking to reporters on a bridge linking Hungary and Slovakia [AFP]
Fico said that the Slovak foreign ministry had handed a diplomatic note to the Hungarian ambassador saying Slovakia would not let Solyom in.
"If he decides to ignore the note it would be a violation of international law and Slovakia's sovereignty," Fico warned.
"We won't prevent him from crossing the border by force. We want to behave in a diplomatic way," he added.
Hungarians make up 10 per cent of Slovakia's population of 5.4 million people, mostly living along the frontier.
Hungary protested to the United Nations over Slovakia's recently-implemented language law, which it dubs discriminatory to the Hungarian minority - a move blasted by Slovakia's leaders.
The law hands fines of up to €5,000 ($7,000) on the use of minority languages in government and other public services.
Ties between the two ex-Soviet bloc neighbours, both of whom joined the European Union in 2004, have been tense since 2006 when the far-right SNS party, known for its antipathy towards minorities, entered the government steered by Fico's left-wing populist Smer party.
Both Hungary and Slovakia are also members of Europe's Schengen zone, where frontier controls have been dropped to smooth cross-border travel.