"What I will not do is sit down with people who are breaking the law," Alward said Thursday after giving a speech at an energy conference in Saint John.
"People have the right to protest, they have the right to raise their concerns, but they don't have the right to break the law."
The protesters have not been charged with any offence. But Alward said it is against the law for them to be blocking a highway and preventing energy company SWN Resources from accessing their property.
The protesters have cut down trees across Highway 134 near Rexton and have blocked the entrance to a compound used by SWN Resources to store equipment for shale gas exploration. They have been demonstrating at the site since Sunday.
The RCMP has been monitoring the protest.
The protesters say they want a meeting with Alward and SWN officials. They are also calling on the government to halt shale gas exploration and for SWN to leave the province.
Opponents of the shale gas industry say the process used to extract the resource — hydraulic fracturing, also known as fracking — could pollute drinking water. But proponents of the industry say such concerns are overblown and don't take into account the possibility of replacing coal with cleaner burning natural gas.