The government had tried to push them through under urgency in March, claiming public consultation had already been done by a previous government.
The bills include giving the government control over key judicial and police appointments, rather than the King.
An opposition MP, Lord Fusitu'a, says the government was strong armed into consultation which showed a strong backing for the status quo.
"It's an overarching statement by an overwhelmingly large majority," he said. "The people have spoken and have rejected six bills, including the constitutional amendment."