There is no obvious spirituality in "To Celia," but the speaker does talk about the soul and implies that there is something spiritual and divine about love. He suggests that a show of love from Celia would be a "drink divine," even more "divine" than the nectar, the drink of the immortal Greek and Roman gods. While the speaker is all about elevating the spiritual over the material, and the "drink divine" of love over actual beverages, he also can't separate spiritual from material; he can't talk about one without the other.
The interweaving of the material (wine, drinking,) with the immaterial (the soul, love) in this poem suggests that, perhaps, the two aren't mutually exclusive alternatives.
The speaker's emphasis on things like love and desire gives way at the end of the poem to an emphasis on more mundane concerns, like how the wreath smells, the fact that it still grows, etc. In other words, the poem seems to back out of its emphasis on the spiritual, on the "thirst that from the soul doth rise" (5).