ARCH (Action on Rights for Children) says schools should also be labelled "failing" if they cannot keep children safe.
Terry Dowty, policy officer for ARCH, said: "A failure to tackle bullying should carry the same penalty as poor academic achievement."
Last week a leading MP tabled an amendment to the Children's Bill which could lead to prosecutions if teachers and local education authorities failed to act on bullying.
But Rhys Williams, of the National Union of Teachers Cymru, said a knee-jerk reaction to the issue was dangerous.
"Teachers are already aware that it is not just their job to teach but also to have regard for the welfare of children. I don't think threatening to use the law against them would make any difference," he said.
Jane Davidson, minister for education and lifelong learning, also remains to be convinced that extra legislation is needed and said the increased bureacracy in setting up tribunals could be counter-productive.
"I don't believe that increasing the level of punitive measures against schools is necessarily in the best interests of young people," she said.
Calls for stronger measures have grown since the death of Neath teenager Laura Rhodes, who took a drugs overdose in an apparent suicide pact with her best friend. Thirteen-year-old Laura left a letter describing how her life had been made a misery by bullies who had taunted her about her weight.
Pupils starting the new term at Laura's former school, Cefn Saeson, in Neath, have all been issued with advice on what to do if they fall victim to cruel bullies.
Tim Pearson, 11, said: "All the Year 7 pupils were told by our teachers that if there is a problem we can go to the head of year. There are also specially-trained Y11 pupils who can act as counsellors to help younger children with problems.
"I think that's a good thing because they were once in our position, so they perhaps understand things better. And it's an alternative to going to a teacher."
Peer support systems were endorsed by NSPCC education adviser, Jane Harries, at a conference on children's mental health held last week in Cwmbran (see above, left).
She said: "They provide opportunities for another young person to share their concerns, to be taken seriously and to explore their situations."