Activists say at least 100 people have been killed over the past two days as troops intensified operations.
Earlier, Turkey said it may need UN help after a surge in refugee arrivals.
After speaking to Mr Ban, Foreign Minister Ahmet Davutoglu said the rate of refugees had doubled since Mr Assad agreed to implementing the UN-Arab League plan.
The six-point peace plan, mediated by envoy and former UN chief Kofi Annan, envisages cessation of armed violence by all parties from 10 April, with a full ceasefire on 12 April.
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Annan’s six-point peace plan
1. Syrian-led political process to address the aspirations and concerns of the Syrian people
2. UN-supervised cessation of armed violence in all its forms by all parties to protect civilians
3. All parties to ensure provision of humanitarian assistance to all areas affected by the fighting, and implement a daily two-hour humanitarian pause
4. Authorities to intensify the pace and scale of release of arbitrarily detained persons
5. Authorities to ensure freedom of movement throughout the country for journalists
6. Authorities to respect freedom of association and the right to demonstrate peacefully
However, the Syrian opposition, the US and many regional countries have expressed scepticism that Syria is committed to the plan.
In Friday’s statement, Mr Ban said the recent attacks on civilians “violate” the UN’s demands and demanded the government halt military operations.
Mr Ban “deplores the assault by the Syrian authorities against innocent civilians, including women and children, despite the commitments by the Government of Syria to cease all use of heavy weapons in population centers,” it said.
“The 10 April timeline to fulfill the Government’s implementation of its (ceasefire) commitments, as endorsed by the Security Council, is not an excuse for continued killing,” it said.
Earlier, Turkey said that if the influx of refugees continued at the current rate, it would need international assistance. More than 2,800 Syrians have crossed over the border in 36 hours, with the total now near 24,000, Mr Davutoglu said.
Many of the refugees have reported intense bombardment by government forces.
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10 April: Government must withdraw troops and heavy weapons such as tanks from towns, cities and villages
Following 48 hours: Ceasefire to be implemented on the ground with the onus on the opposition to follow the government’s lead
06:00 local time on 12 April: All forms of violence must be stopped on all sides
Next step: All parties to hold talks on a political solution
On Friday, activists said violence was continuing across the country.
The BBC’s Jim Muir in Beirut says that in Homs and some other areas the approach of the deadline to halt all violence seems to have brought an escalation rather than a reduction.
He says activists are accusing the government of trying to complete its crackdown come what may; but government officials say it is the rebel fighters who are exploiting the impending withdrawal of the military from towns and cities.
A UN team is currently in Damascus to negotiate the possibility of deploying UN monitors to oversee any ceasefire.
Mr Annan has said that if the truce is successful a small mobile UN monitoring mission of some 200-250 observers could be brought into Syria.
The UN estimates more than 9,000 people have died in the year-long uprising against the rule of President Assad.