By David Winning

SYDNEY--Rio Tinto PLC should authorize an independent and transparent review of all agreements negotiated with indigenous groups, a pension-fund investor said Thursday, as pressure builds on the miner for a stronger response to the destruction of two ancient rock shelters in Australia.

Hesta, an Australian pension fund for health-care workers, said it had lost confidence in Rio Tinto's ability to address issues around its agreements with traditional landowners on its own. Those issues weren't addressed by Rio Tinto's review into the May 24 destruction of two caves at Juukan Gorge in the minerals-rich Pilbara region that were culturally significant to a local indigenous group, it said.

Rio Tinto has apologized for blowing up the caves during work to expand its iron-ore operations, and last month cut bonuses for Chief Executive Jean-Sébastien Jacques and two other executives for governance failures outlined in an internal review that aimed to prevent similar missteps from happening again.

In a review of the event released Aug. 24, Rio Tinto found that no single individual or error was responsible for the destruction of the rock shelters that were significant to the indigenous group, the Puutu Kunti Kurrama and Pinikura people, or PKKP.

"Accountability for the destruction at Juukan Gorge should rest at the highest levels of Rio but a larger, systemic issue of how the company and the mining sector negotiates agreements with traditional owners needs to be urgently addressed," Hesta said in a statement.

Hesta, which owns Rio Tinto shares worth around $182 million, said Rio Tinto's public statements to a federal parliamentary inquiry into the caves' destruction indicated that agreements with traditional owners exist, "requiring the surrender of certain legal rights, past and future, and they remain currently in use."

Without an independent review, Hesta said it couldn't adequately assess the reputational and legal risks to Rio Tinto and the impact this could have on the company's value.

Rio Tinto declined to comment on Hesta's statement.

Write to David Winning at david.winning@wsj.com