The meter of the song is "Irregular with refrain", which in practice works out to be:

78.65.65.65
8.10.8.10

Really, the first couple lines are 65, just with extra grace notes stuffed in. You can stuff in or pull out notes pretty much at will, because folk tunes are good for that.

There's some kind of plain old 65.65 St. Alphonse Liguori song I found on Cyberhymnal, of all places, that stuffs well, although the tune really doesn't have enough gravitas unless you do that big striding thing with it that Appalachian Spring did. It's a Caswall translation done for the Birmingham Oratory, so it's sure enough Catholic. I don't think this is what you want for a wedding, necessarily, but it's an example of what you could do. You'll notice that each verse is two 65 verses stuffed together, whereas the refrain is one 65 verse that's been stretched to fit. So there's one quick and dirty way to mung something together. (Another thing you might do is steal the verses from a 65 hymn and the refrain from something more like 8.10.)

Blest through endless ages Be the precious stream
Which from endless torments Doth the world redeem.
Grace and life eternal In that blood I find;
Blest be His compassion, Infinitely kind.

REF:
Glory be to Je-e-e-esus,
Who, in bitter pains,
Poured for me the li-i-i-ifeblood
From His sa-a-acred veins!

There the fainting spirit Drinks of life her fill;
There as in a fountain Laves herself at will.
Abel’s blood for vengeance Pleaded to the skies;
But the blood of Jesus For our pardon cries.

Oft as it is sprinkled On our guilty hearts,
Satan in confusion Terrorstruck departs.
Oft as earth exulting Wafts its praise on high,
Angel hosts, rejoicing, Make their glad reply.

Lift we then our voices, Swell the mighty flood;
Louder still and louder Praise the precious blood!
Lift we then our voices, Swell the mighty flood;
Louder still and louder Praise the precious blood!

How about just playing a prelude or postlude based on the tune? (As a recessional, maybe?) I think it's too jiggy to be the basis for a good hymn, even with ideal Catholic words.

You could use this translation of the Te Deum for the verses, perhaps repeating one verse as the refrain.

I.e.

1 Thee, God, we praise, Thy name we bless,
Thee, Lord, of all we do confess;
The whole creation worships Thee,
The Father of eternity.

R: O God eternal, mighty King,
We unto Thee our praises bring;
And to... Thy... true and only Son,
And Holy Spirit, Three in One.

2 To Thee aloud all angels cry,
The heav'ns and all the pow'rs on high,
The cherubs and the seraphs join,
And thus they hymn Thy praise divine.

R: O God eternal, mighty King,
We unto Thee our praises bring;
And to... Thy... true and only Son,
And Holy Spirit, Three in One.

3 Holy, holy, holy Lord,
Thou, God, of hosts, by all adored;
Earth and the heav'ns are full of Thee,
Thy light, thy pow'r, Thy majesty.

(etc).

Or, sing a blessing on the couple and their new home? This could work, with a little tweaking.

I like Rebecca's ideas....

Here's another one. I guess it's not a hymn, per se.

When the Lord God looked upon the man as he roamed,
He said, "It's not good for man to be alone.
I will make him a helper who can stand by his side,
To be a fitting match and be his bride."
She was not made out of his head
Nor from his feet, at the very start
But from his rib, so that when they were wed,
She'd stand by his side, closest to his heart.

When the man woke to see her, and no more was alone,
He said, "At last! She's bone of my own bone,
Flesh of my own flesh from so close to my heart,
That we can never really come apart."
A man leaves home and he cleaves to his wife,
And what God has joined, let none pull apart.
Since as one flesh, they make a brand new life,
May they stand together and be of one heart.

Or you could do marriage blessings from the Bible, like the prayer from the Book of Tobit or the psalm:

Lord God of our fathers, may the heavens and earth,
Oceans, rivers, and the springs that gave them birth,
And all of Thy creatures in the world, bless Thee
And may this wedding be to Thy glory.
For Thou madest Adam from the clay,
And Eve is forever his help and stay
And from them came all of mankind.
Let them grow old together, Lord so kind.

Blest are all those who fear the Lord and walk in his ways.
You shall eat your labor's fruit all of your days.
You will have blessings and prosperity
Like a grapevine you'll live fruitfully.
Like branches of the olive tree
Around your table your children be.
Your children's children may both of you see,
Those who fear the Lord so blest will be.