King Salman was announced as the highly conservative kingdom's new ruler overnight, after the death of King Abdullah, his half-brother.
New York-based HRW urged him to make real reforms in the kingdom, which is frequently criticised by rights groups and the international community.
"King Abdullah came to power promising reforms, but his agenda fell far short of achieving lasting institutional gains for Saudi citizens," said deputy Middle East director Joe Stork.
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"King Salman... should move the country forward by ending intolerance for free expression, rooting out gender and sectarian discrimination and fostering a fair and impartial judicial system."
HRW acknowledged that King Abdullah, generally considered a modest reformer by the kingdom's standards, had pushed for changes early in his rule.
But it said that, after 2011, "authorities subordinated the king's reform agenda to a campaign to silence peaceful dissidents."
In particular, HRW urged King Salman to "prohibit discrimination against women and religious minorities and institute protections for free speech."