In a video comment posted on the presidential website, Poroshenko called the law "part of our strategy towards fully breaking with the colonial past and reorientation towards Europe."
The treaty is due to expire on March 31. On December 6, Ukrainian lawmakers voted not to prolong it beyond that date.
Signed in 1997, the treaty obliges Russia and Ukraine to "respect the territorial integrity of each other and confirm the inviolability of current mutual borders."
It also says that Ukraine and Russia should build bilateral relations "based on principles of mutual respect of sovereign equality, inviolability of borders, peaceful resolution of differences, without the use of force or the threat to use force."
Ukrainian government forces have been fighting against Russia-backed separatists in the eastern Ukraine since April 2014, shortly after Russia seized Ukraine's Crimean Peninsula and forcibly annexed it.
Although Moscow denies interfering in Ukraine's domestic affairs, the International Criminal Court (ICC) in November 2016 ruled that the fighting in eastern Ukraine is "an international armed conflict between Ukraine and the Russian Federation."