That's not exactly the kind of swearing Abraham had in mind when he asked his servant to put his hand under his thigh and swear by the Lord. He was actually getting his servant to promise that he'd go find a wife for his son from the land he came from, rather than the place where he now lived. He was getting old in years and expected to die soon. He didn't want a Canaanite woman for his daughter-in-law. And he didn't want his son to move back to where he used to live. God had given him and his offspring the land of Canaan, and moving away would be like giving God back His gift or something. He didn't even want his son to come along for this important wife-finding mission.
This story is quite funny, and profound as well. The servant went to the town of Nahor. He had the ten camels (which were loaded with "all kinds of good things") and waited. He prayed, "O Lord, God of my master Abraham, give me success today, and show kindness to my master Abraham. See, I am standing beside this spring, and the daughters of the townspeople are coming out to draw water. May it be that when I say to a girl, 'Please let down your jar that I may have a drink,' and she says, 'Drink, and I'll water your camels too' -- let her be the one you have chosen for your servant Isaac. By this I will know that you have shown kindness to my master."
The text says that "before he had finished praying, Rebekah came out with her jar on her shoulder. She was the daughter of Bethuel son of Milcah, who was the wife of Abraham's brother Najor. The girl was very beautiful, a virgin; no man had ever lain with her. She went down to the spring, filled her jar and came up again." It says: "The servant hurried to meet her." Then he went into his spiel about the water.
You better believed he hurried! When it says "she was very beautiful," it means what it says. This servant could have been shaking and been very nervous and excited about finding such a knockout babe for Isaac to marry. He also could have been pouncing on the idea that she was obviously God's answer to his prayer, since she appeared instantly -- even before he was done praying.
He asked her, "Please give me a little water from your jar." She didn't look at him like he was weird or tell him, "Hey, I came all the way down here with this heavy clay pot to fetch some water for my household. Why don't you go down into the spring and get your own water?" Instead she said, "Drink, my lord," and quickly lowered the har to her hands and gave him a drink. After she had given him a drink (which the servant must've been silently and excitedly waiting to find out what happened next) she said, "I'll draw water for your camels too, until they have finished drinking." She emptied the jar into a trough for the camels to drink and ran back to the well to draw some more, getting enough for all his camels. She not only wet the camels' tongues, she committed to watering them until they'd had their fill. This might've taken a few jarloads of water for ten camels...
When the camels were done drinking, he took out a gold nose ring (that's right) and two gold bracelets. He asked her who her parents were and, in the same breath, asked if he could stay the night. When he heard that they were Abraham's relatives, he was overjoyed. She affirmed that they had enough straw and fodder for all the camels, as well as room for him.
When he got there they told him they'd prepared a meal for him. He said, "I will not eat until I have told you what I have to say." He laid out the story of Abraham's blessings and prosperity and the oath he swore. This oath had an "out," by the way, which let the servant off the hook if the woman refused to come back with him. He explained this "out" clause to them as well. He explained the prayer he said to the Lord and stated that: "Before I finished praying in my heart, Rebekah came out..." This is biblical text that shows that praying silently is valid, real, and biblical. It's always nice to know some of these practical things. Laban and Bethuel told the servant that they wouldn't refuse the Lord and they let her go marry Isaac. The next morning he got ready to go, but a brother and mother of Rebekah tried to talk him into waiting ten days. The servant wisely refused, knowing that giving into this delay might turn into a complete failure of "the mission." They said, "Let's call the girl and ask her about it." When they did, she said, "I will go." So they mounted the many camels and went back to Canaan. Rebekah's nurse (who is mentioned 7 chapters later when she dies) and her maids all went with. It was smart to bring the ten camels, that's for sure.
When they got close they ran into Isaac. Isaac saw the camels approaching. Rebekah looked up and saw Isaac. She asked who it was and when she found out she covered herself with her veil. They got married and he loved her.
I heard a beautiful analogy about this story once:
Abraham is like God our Father.
Isaac is like Jesus the Son.
Abraham's servant is like the Holy Spirit, Who has all the gifts of the Father with Him, which He uses to woo a bride for the Son.
It makes sense, doesn't it?