Ukraine conflict: US condemns Russia for 'provocative' actions
The warning comes hours after Ukraine said itsforces attacked and partly destroyed a Russian armoured column that entered its territory.
The Russian government denied its forces crossed the border and accused Kiev of trying to sabotage deliveries of aid.
Kiev suspects the aid convoy destined for besieged rebel-held cities could be a "Trojan horse" to provide military help to insurgents.
White House national security council spokeswoman Caitlin Hayden said there had been an "escalation" in Russian activity in recent weeks to unsettle the already tense situation on the ground in eastern Ukraine.
"Even as we work to gather information (about the partial destruction of the Russian armoured column), we reiterate our concern about repeated Russian and Russian-supported incursions into Ukraine," Ms Hayden said.
"Russia has no right to send vehicles, persons, or cargo of any kind into Ukraine, under any pretext, without the government of Ukraine's permission."
Russia firing rockets into Ukraine: US
Ms Hayden said Russia has also been firing artillery and rockets from Russian territory into Ukraine "on a regular basis".
"The escalation in Russian activity designed to destabilise Ukraine in recent weeks is extremely dangerous and provocative.
"It includes supplying separatist fighters with tanks, armoured vehicles, artillery and multiple rocket launchers."
Moscow has consistently denied supporting rebels in eastern Ukraine and the crisis has led to tit-for-tat sanctions between the West and Russia over Moscow's alleged behaviour.
NATO separately also accused Russia of active involvement in the "destabilisation" of eastern Ukraine, where pro-Kremlin separatists have been fighting against Kiev for four months.
Ms Hayden said Russian president Vladimir Putin can help put a stop to the conflict "by ending Russian firing into Ukraine; stopping its supply of weapons, support and cash to separatists; and closing the border to militants, including by facilitating an effective border monitoring mission under the auspices of the OSCE".
"The continued Russian military intervention into Ukraine is entirely at odds with legitimate efforts to provide humanitarian assistance and negotiate a peaceful resolution to the overall crisis."
No troops part of 'humanitarian convoy', Russia says
Earlier Ukraine said its border guards and customs officials had begun inspecting a Russian humanitarian aid convoy bound for eastern Ukraine.
A total of 280 Russian trucks have brought the aid to the border.
Russia said the convoy of white-painted trucks was carrying 2,000 tonnes of water, baby food and other aid for people in the region.
There have been allegations the drivers dressed in khaki "uniforms" are soldiers.
US defence secretary Chuck Hagel called his Russian counterpart Sergei Shoigu on Friday and was promised that no Russian troops are part of the "humanitarian convoy".
"Minister Shoigu 'guaranteed' that there were no Russian military personnel involved in the humanitarian convoy, nor was the convoy to be used as a pretext to further intervene in Ukraine," the Pentagon said.
The Pentagon statement said Mr Shoigu had insisted to Mr Hagel the convoy was not a disguised military column and vowed the aid would be properly distributed by a neutral third party.
"Minister Shoigu assured Secretary Hagel that Russia was meeting Ukraine's conditions," it said.
"He acknowledged that the goods would be delivered and distributed under the International Committee of the Red Cross," it added.
"The two ministers discussed the need to have bilateral follow-on meetings of the Intermediate-Range Nuclear Forces Treaty. They also agreed to maintain open lines of communication."