Yes, there is a scriptural reference from the Ramayana. In fact, Chinna Jeeyar Swami has himself given the reference where to find. Swamiji said "After killing Ravana", so it is found in the Yuddha Kanda Sarga 117 and 119 in the Valmiki Ramayana.
From Sarga 117 of Yuddha Kanda,
ततो वैश्रवणो राजा यमश्च पृभिः सह |
सहस्राक्षश्च देवेशो वरुणश्च जलेश्वरः || ६-११७-२
षड्र्धनयनः श्रीमान् महादेवो वृषध्वजः |
कर्ता सर्वस्य लोकस्य ब्रह्मा ब्रह्मविदां वरः || ६-११७-३
एते सर्वे समागम्य विमानैः सूर्यसंनिभैः |
आगम्य नगरीं लङ्कामभिजग्मुश्च राघवम् || ६-११७-४
Thereupon, Kubera the King of Yakshas, Yama the lord of death together with the deceased ancestors, Indra the lord of celestials Varuna the lord of waters,** the illustrious Shiva the great deity who bears the device of a bull as his banner and having three eyes**, Brahma the creator of all the worlds and the best among the knowers of sacred knowledge all these together reaching the City of Lanka in aerial cars, shining like the sun approached Rama. [6-117-2]
ततः सहस्ताभरणान् प्रगृह्य विपुलान् भुजान् |
अब्रुवंस्त्रिदशश्रेष्ठा राघवं प्राञ्जलिं स्थितम् || ६-११७-५
Lifting their long arms, their hands decked with ornaments, those excellent gods thereupon, spoke (as follows) to Rama who stood there, making a respectful salutation to them with his folded hands. [6-117-5]
So all the devatas appeared before Sri Rama after Ravana was killed and agni Pariksha was done for Sita.
For the statement of Shiva praising Sri Rama:
Chinna Jeeyar swami is probably referring to this statement where Shiva praises as the saviour of the wise and punisher of the wicked. From Sarga 119,
पुष्कराक्ष महाबाहो महावक्षः परन्तप |
दिष्ट्या कृतमिदं कर्म त्वया शस्त्रभृतां वर || ६-११९-२
O lotus-eyed, long-armed, broad-chested, annihilator of enemies and excellent among those upholding the cause of virtue! Thank heaven! You accomplished this task. [6-119-2]
Ravana is indeed wicked. So, this statement is also right.
So, statement has scriptural basis. As a suggestion, the complete article is worth reading where Swamiji says that one can worship Lord Shiva and should not spread wrong stories about incarnation of Lord Vishnu.