There are plenty of poems that you could look at in order to answer this question. Whitman is of course recognised as being one of the most famous American Romantics, and so it follows that the vast majority of his works bear this stamp of being Romantic works of art.
One of my favourite Romantic poems, however, is entitled "When I Heard the Learn'd Astronomer." This poem is Romantic because of the contrast that is created through comparing a view of the night sky and the stars and planets in it that is strictly rational and based on "charts," "diagrams" and dividing and measuring them with a view that is much more mystical and instrinsic. The poem presents the speaker as attending a lecture on the universe by the "learn'd astronomer" who is clearly an expert in his field. What is strange however is how this lecture leaves the speaker feeling ill-at-ease and rather depressed. It is only when he goes outside and looks at the night sky from his non-expert point of view that he feels better:
I wander'd off by myself,
In the mystical moist night-air, and from time to time,
Look'd up in perfect silence at the stars.
A key element of Romanticism is the way that it depends on intuition and feelings rather than on scientific formulaes and equations. These two aspects are contrasted in this poem, and the way in which the speaker finds peace and tranquility in the night sky by himself indicates the Romantic force of this poem.
The Romantic movement privileged intense human emotion and experience over logic and reason, as they believed that our emotions are more authentically "us"; we do not have to be taught to feel strongly. On the other hand, we must cultivate logic—it is not as immediately knowable to us as emotion. We see this prioritizing in "What think You I take my Pen in Hand?" In this poem, the speaker discusses what he sees that makes him pick up his pen to write: it is a personal interaction between two men while they are preparing to part from one another. It is an emotional scene as the men clearly feel very deeply attached to each other:
The one to remain hung on the other's neck, and passionately kiss'ed him, / While the one to depart, tightly prest the one to remain in his arms.
The fact that it is this moment of intense emotion that so arrests the speaker's attention helps to position poem this firmly in the Romantic tradition. The sight of these two men experiencing such intense emotion beats out a "perfect-model'd, majestic" battle ship, as well as "The splendors of the past day" as well as night, and the "great city spread around" the speaker. Instead of any of these "majestic," "splen[did]," and "great" sights, he is most affected by the deep display of passion he witnesses.