From a Judeo-Christian religious perspective, God can be said to be the "owner" of the woods to which the poet refers, but Frost explicitly identifies the wood's owner:
His house is in the village, though;/He will not see me stopping here/To watch his woods fill up with snow. (ll. 2-4)
One can argue, especially in "Stopping by Woods on a Snowy Evening," that a contemplation of nature, rather than God, is one of the central themes of the poem, and that Frost means for us to focus our attention on the contrast between the poet's reverie and enjoyment of nature, on one hand, and his obligations to move on, on the other hand.
The poet, who has stopped to watch the woods "fill up with snow," immediately realizes that his horse does not have the same reaction to the pause as the poet does:
My little horse must think it queer/To stop without a farmhouse near . . . The darkest evening of the year. (ll. 5-8)
The horse, a beast of burden not given to enjoying the silent falling of snow and most certainly not in the midst of a reverie, simply doesn't understand a stop in the middle of some dark woods with no destination (comfort) in sight. The man and horse have contrasting perceptions of their surroundings because they have vastly different goals--the man, a spiritual goal; the horse, a physical goal of oats and a warm stall.
In the third stanza, this contrast between man and beast becomes even more explicit when the horse
. . . give his harness bells a shake/To ask if there is some mistake. (ll. 10-11)
The horse, oblivious to the beautiful solitude and silence, is simply reminding the poet that it is time to move on. The poet, still deep in his reverie, notes that he hears only "the sweep/of easy wind and downy flake. Again, horse and man inhabit different worlds--the beauty and solitude of nature are lost on the horse but capture the man.
The final stanza pulls the man out of his reverie by reminding him that
. . . I have promises to keep,/And miles to go before I sleep,/And miles to go before I sleep.
The poem's last two lines likely have a double meaning: the poet has immediate obligations to perform before he can sleep that particular night and, more important, he has extended, lifelong obligations to meet before he takes his final sleep.