The regulator said, external the site had blocked access for the "vast majority" of UK users, and it had "no further legal routes available" under the Online Safety Act.
But it accepted people would still be able to access the forum by using a VPN, which hides your location online.
Andy Burrows, chief executive of the Molly Rose Foundation, said the decision to take no further legal action would "result in more lost lives" and that announcing it on the day a new prime minister was named showed "an utter lack of moral courage".
In response, an Ofcom spokesperson said it had made the decision a "matter of days ago" following "months of work to exhaust all possible legal routes".
They said bereaved families were informed of the decision this morning after being contacted last week.
"In that context, we do not consider it appropriate to delay the announcement further to accommodate today's political developments," they said.
Meanwhile, Ofcom enforcement director Suzanne Cater said the regulator had used its powers "to the fullest extent possible".
"We'll continue to monitor the forum closely to check the geoblock remains in place," she said.
The geoblock means people based in the UK are unable to access the site, unless they mask their location through a VPN or similar means.
Ofcom said it was working with the government to explore whether its enforcement powers should be strengthened.
In particular, this concerns its ability to get a website blocking order through the courts, which would legally mandate internet service providers in the UK to block the website.
The regulator says it has reached the limit of what it can legally do under the Online Safety Act because the forum has implemented the geoblock.
The BBC has approached the government for comment.
The forum, which the BBC has chosen not to name due to its nature, was the first service to be investigated by the regulator under the UK's online safety laws.
BBC News investigations into the forum found young people who had been on the site had been encouraged to and given instructions on how to take their own lives.
It has been cited in multiple coroners' reports regarding the deaths of UK citizens.
In May Ofcom issued a £950,000 fine to the site, saying it did not comply with the Online Safety Act "to protect people in the UK from illegal content", and warned it was preparing to seek a court order to block access if compliance issues were not addressed.
The firm had until 12 June to pay the fine. Ofcom said it has not paid.
If a company fails to pay a fine, Ofcom can still try and recover the debt through UK-based assets - if the company has any - or through a more complex process of engaging enforcement agencies in other countries if it does not.
Sarah Ruane, director of policy at suicide prevention charity Samaritans, said it was "extremely disappointed" by the announcement.
"After months of investigating this suicide forum, and over three years since the Online Safety Act was passed, we are yet to see meaningful action against some of the most harmful sites online," she said.
"All today's announcement does is raise serious questions about whether the Online Safety Act is fit for purpose."
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