The answer to this is very simple and appears directly stated in the fourth and final stanza of the poem.

A God, a God their severance ruled!
And bade betwixt their shores to be
The unplumb'd, salt, estranging sea.

Having said that, you probably see the answer for yourself now, but let's build up to it anyway. First, the poetic speaker, the persona, is Matthew Arnold himself. Since he is the speaker, he is not likely to blame himself for separation from what he feels "longing like despair" for.

Let's ask the question: What brought the island together? The answer is in the dramatic second stanza. It was the moonlight on island hollows, balmy spring winds, starry nights, the nightingales' songs pouring forth that swept the islands together as the "lovely notes" bounced "from shore to shore":

And lovely notes, from shore to shore,
Across the sounds and channels pour--

Now let's ask: What did the isolated islands feel they were? Stanza three answers this by defining the source of the "longing like despair." Though now separated, with separate boundaries, or margins ("marges"), they felt they were once united as parts of one whole, one metaphoric "continent":

For surely once, they feel, we were
Parts of a single continent!  

Now, who separated them becomes more clear. It was neither of them. It was no force of nature. It was the providence of God, who set the laws of individuality in place.

The islands feel the enclasping flow,
And then their endless bounds they know.

God's providence might also refer to the deep discord between Protestant and Catholic  (Arnold was Protestant Anglican while Marguerite is thought to have been Catholic). This discord in theology was perceived as producing a great divide between practitioners of each: they could not be united because their religions divided them.