What evidence does the speaker give of the depth and the constancy of his love?
Ah...Robert Graves! He wrote poetry for the oldest and possibly the most respected reason in the book: to woo women. He wrote poetry / songs to play at the pub.
He was Irish, so some of the words are a bit odd to modern Americans. "Gang," for example, means "go," and "weel" is "well." He uses similes and analogies to emphasize the depth and endurance of his love.
His love is "like a red, red rose / That's newly sprung in June" shows through the color and feel of spring the depth and promise of his love. Red is a deep, rich color--the color of love itself.
He will love her until all "the seas gang dry," an analogy that shows he will never stop loving her, as we can reasonably expect that all the seas will never go dry. Even the Dead Sea, which has been drying up steadily for centuries (and has no outlet, which explains the intensity of its salt content) will not dry up for many centuries yet, far longer than any human life.
He goes on to say he'll love her until "all the rocks melt wi' the sun," another impossibility, as rocks only melt with the intense heat of the earth's core. He emphasizes the constancy of his love with the analogy "I will love thee still, my dear, / While the sands of life shall run." This uses the metaphor of an hourglass, but this hourglass is big enough to encapsulate the length of life itself, and it will never stop running.
In the end, he says goodbye to her ("fare thee weel"), and adds that he will come again to her "Thou’ it were ten thousand mile." Nothing, he says, will keep him from her.