"What have they been there doing? What solution have they brought to the problem? What is the result of the $1bn they are spending every year?" he told the BBC.
He said the real reason for the crisis was what he described as the weak leadership of the Congolese government, which had failed to disarm Hutu rebels.
Mr Kagame is due to meet his Congolese counterpart Joseph Kabila at a summit on Friday in Kenya to be attended by UN chief Ban Ki-moon.
The UN mission, Monuc, is reinforcing its troops in the regional capital of North Kivu, Goma, and has warned that they will fire on any armed group trying to enter the city.
In the latest fighting, Gen Nkunda's CNDP entered and took control of the centre of the large town of Nyanzale, about 80km (50 miles) north-west of Goma, Monuc spokesman Lt-Col Dietrich said.
Gen Nkunda defends his military actions
During the fighting, the Congolese army evacuated their 15th brigade headquarters, he said, adding that the rebels were "breaking their own declared ceasefire".
"It's clear they are trying to have a territory completely under their control," he told the Reuters news agency.
Earlier, the rebels ordered thousands of people to leave their homes as they re-took control of the town of Kiwanja.
Witnesses reported killing and looting, and some people were wounded, a BBC correspondent who travelled to the town said.
Gen Nkunda told the BBC that residents had been told to move for their own safety so his troops could fight Pareco Mai-Mai forces, who he claims are backed by the Congolese government. He said people were now being allowed to return home.
"Mai-Mai were in some houses in Kiwanja - it was difficult for our troops to fight where there is so many people, so they asked them to be out of the fire," he said.
The recent fighting has forced tens of thousands of people to travel south to Goma, which Gen Nkunda has threatened to attack - though a ceasefire around the city appears to be holding for now.
UN soldiers and journalists found at least a dozen corpses in Kiwanja
The offensive has also prompted some aid workers to suspend their activities a day after bringing in the first food convoy to rebel-held territory.
The UN refugee agency has said three camps for displaced people near the town Rutshuru have been destroyed, and that it is trying to establish the fate of about 50,000 people who had been there.
Gen Nkunda has threatened to topple the DR Congo government in Kinshasa, 1,580km (980 miles) west of Goma, unless President Kabila agrees to hold direct talks.
He has said his forces are now free to pursue their offensive, accusing the government of breaking the ceasefire.
Correspondents say the militia involvement in the fighting makes any push for negotiations between Gen Nkunda and the government more complicated.
The latest clashes sparked fears the rebels could follow through on their threat to attack Goma.
The BBC's Peter Greste in Goma says the rebel threat against Kinshasa could be hubris, as it is hard to see how he could transport between 6,000 and 7,000 fighters all the way across a country the size of western Europe.
The rebel forces do, however, appear to have the strength to take Goma, he says.
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