Valmiki Ramayana does not specify to the fact directly.
But it states Rama worshipped Shiva at Rameswaram.
That Rama installed a Shiva lingam is referred to in Adhyatma Ramayana,Shiva Purana,Linga Purana,Skanda Purana, Padma Purana,Ramacharitamanas by Tulsidas and Shiva Daridra
Here I make an exception in producing evidence as this blog is considered authentic with references and is being quoted in research papers and Wiki.
I owe them this much.
Lord Rama,while returning to Ayodhya after the war at Sri Lanka by Pushpaka Vimana describes the places he had visited to Sita.
He describes Rameswaram as the place where Shiva showered His Blessings on Him( Rama)
And more.Please check the related posts in each article mentioned above.
I have also written on the Pyramids of Egypt, Temples in the South East and Middle East.
There seems to be some geometry associated with them, which we are yet to unfathom.
I am checking on these.
I am producing information on Eight Shiva Temples of Shiva.
They are located in the same Longitude,
It is the belief Hindus that Rama worshiped Shiva at Rameswaram.
And Rama built a bridge from here to Lanka.
What do Valmiki and Vyasa say on this?
Anjaneya, Hanuman is considered to be an Avatar of Shiva and some Purans mention Him as the Son of Lord Shiva.
Though Hanuman was an ardent Devotee of Lord Rama, He was a devotee of Shiva as well.
But there is an incident narrated in the Valmiki Ramayana, where Hanuman uprooted the Shiva Linga!