In April, the military alliance decided not to offer Ukraine and Georgia a membership action plan, a path to joining Nato, but agreed to review the situation in December.
During a visit to Slovenia on Wednesday, the Ukrainian president said: "We decided to take the road of approaching Europe and Nato, this is our strategic goal, our foreign policy, it is not a policy against some third country, against any neighbour.
"From a political point of view, it would be a big mistake if [Russia] now prepared a kind of revision of that treaty.
"I will not allow any possibility of raising the voice or provocations [against Russia]. I will do nothing that could throw a wrong light on Ukraine or worsen our relations.
The president said Ukraine and Russia currently had friendly relations set "at a high level."
Earlier, Yulia Tymoshenko, Ukraine's prime minister, said "radical steps" would only serve to disrupt relations between the neighbouring countries.
The Duma, Russia's lower house of parliament, voted by 408-5 to adopt the resolution.
The Kremlin often uses the chamber to issue tough resolutions on disputes with its neighbours, a ploy analysts say is designed to strengthen the government's negotiating position.
Scrapping the treaty could, in theory, open the way for Russia to mount a legal challenge to Ukraine's sovereignty over Sevastopol.
The treaty recognises the port as within Ukraine's borders and Kiev says it is part of its territory.
However, Russian legal experts say without this document, the legal grounds for its status as part of Ukraine are weak.
The Crimean peninsula, which includes Sevastopol, was part of the Russian republic of the Soviet Union until 1954, when Nikita Khrushchev, the Soviet leader, signed it over to the Ukrainian republic.
Some in Russia say Khrushchev's decision was illegal.
Yushchenko is due to meet Medvedev later this week at an informal summit of ex-Soviet states.
The talks will be their first since Medvedev took over as president from Vladimir Putin last month.