A literary theory is something intended to give an adequate explanation of a literary text. Thus all critical theories can be applied to Robert Browning's "Love among the Ruins," with different types of literary theory serving as ways to explore different aspects of the text or provide different types of interpretation.
One approach would be New Criticism, which involves close reading of the poem to understand the interplay of literary form with theme. In a New Critical approach, you might look at the poem as a commentary on the genre of the pastoral, and in particular as a response to Romantic poetry which locates the pastoral as rural or pre-industrial by creating a post-Apocalyptic genre of urban pastoral. By alternating between descriptions of the pastoral present and urban and militaristic past, Browning blends many poetic genres underneath his basic pastoral framework. The conclusion that "Love is best" would be read under such a theoretical framework as a comment about love as an important theme of all poetry, whether urban or rural.
Another possibility might be a Marxist reading of the poem as about how the inherent conflicts of capitalist society might result in its inevitable destruction. The treatment of "love" could be seen as a form of false ideology.
A feminist critique would look at how the woman in the poem is not allowed to speak at all, but merely acts as a silent object of the poet's gaze. This is particularly interesting in light of Browning's marriage to Elizabeth who was far more successful as a poet.
A New Historical approach would examine how the poem reflects specific elements of its Victorian context, including the question of Empire, the woman question, and the ways in which love or personal spirituality might replace the collective ideology of religion as a form of solace in extreme situations in light of the rise of Higher Criticism.