what evidence does walton provide of his romantic leanings?

I'm not sure if you mean "learnings" or "yearnings" but in any case, the guy is a hopeless explorer and adventurer. An obvious example is Walton's wonder at the rough beauty of the north. Certainly, in the expression of his letters and in Walton's determination to act as an individual, readers identify the Romantic soul in the man. He first writes of his feelings about the Artic,

Inspirited by this wind of promise, my day dreams become more ferven and vivid, I try in vain to be persuaded that the pole is the seat of frost and desolation; it ever presents itself to my imagination as the region of beauty and delight.

This love of nature permeates Walton's letters as does his desire, his dream, to strike out as an individual. In addition, Walton remarks upon his earlier desire to be a poet, and poetry was regarded by the Romantics as the highest form of literature,

I also became a poet, and for one year lived in a Paradise of my own creation; I imagined that I also might obtain a niche in the temple where the names of Homer and Shakespeare are consecrated.

However, when he failed at poetry, he turned to science and went out on several expeditions to the North Sea where he became enchanted with the Artic.

In his second letter to his sister Margaret, Walton expresses another Romantic idea, the importance of friendship. For the Romantics the friendship between two men was viewed as the most perfect of loves. And, it is for this communication of feeling that Walton yearns.

He does, however, express his feelings well to his sister--poetically, in fact. for he tells his sister that he is going to "unexplored regions, to 'land of mist and snow.'" He alludes to the "Ancient Mariner"; and he explains that there is something at work in his soul which he cannot understand. Yet, he ventures forth,

What can stop the determined heart and resolved will of man?

My swelling heart involuntarily pours itself out thus....

Romantic yearnings propel Walton toward the Artic Sea and adventure.

none