ACT LIKE...= To behave as if..., to do things thinking that...

STAND AND WATCH= This phrase is a reference to on-lookers. When there is an accident or a fire or some other catastrophe, many people gather around and "stand and watch", but do nothing, simply watch (because they can't do anything or because they don't want to do anything).

MOVE ON= Stop what I was doing and do something different.
- I'm tired of watching telly, time to move on. Why don't we go for a walk?
So the sentence "I could say a prayer and just move on" means: I could pray for all the people who were suffering there, and then simply leave all the responsibility in God's hands, forget about it and continue with my life as if nothing had happened.
Other meanings:
When something bad or traumatic happens in your life you may feel deeply affected by it, but sooner or later you must move on. For example, if somebody you love dies, or you and your lover break up, you feel devastated, but you need to process that, heal up and move on (= continue with a normal life and stop thinking so much about it).
- I know how close you felt to your father, but it's been a year now, you must move on.
In colloquial English we also use this expression meaning "you're obsessed with that, just forget about it and think of something else":
- Stop telling me about Susan, man, move on!

LIKE= (conjunction) AS IF
Like nothing's wrong = As if nothing was wrong. (notice like+present = as if+past)

REFUSE= If you refuse, you say NO to it, you don't want that.
It can also be used as a transitive verb:
- I refuse to do it = I don't want to do it (but "refuse" sounds stronger)

CAUSE= (coll.) Because.

EMPTY PRAYER= A useless prayer /preə*/, a prayer which is only words, but nothing more; you simply pray for a problem to be solved but don't get personally involved in it.
To pray is to talk to God (or other beings in the spiritual dimension)

SIT AROUND= To be around (here and there), usually doing nothing in particular and with no specific purpose. So, basically, wasting time.

SOMEONE ELSE= Another person.
We can use ELSE with the compounds of some- any- and no- meaning "another":
- I don't need anything else = I don't need any other thing
- Let's go somewhere else = let's go to a different place (to another place)
- There was nobody else in the room = There were no more people in there

GOD HAS CALLED ME TO DO MYSELF= If God calls you to do something, He planned that you should do it, He wants you to do it. But then you are free to do it or not, that's why he says "I could choose not to move, but I refuse" (= I refuse not to move, I'm going to do something)
 
THE LEAST OF THESE= The weakest people, those who are more in need.
This phrase comes from the Bible, when Jesus said (talking about Judgement's Day) "Truly I tell you, whatever you did for one of the least of these brothers and sisters of mine, you did for me" (Matthew 25:40)

CRYING OUT= Shouting loudly, as when asking for help or moaning about their suffering.

FOLLOW THROUGH= To do something till the end, until it is finished.

THE WEARY AND LOST= People who are tired (in an existential sense) and live life without a meaning, have no sense of direction or purpose.
WEARY /wɪərɪ/ means "tired" (physically and/or mentally). We can use THE + ADJECTIVE to talk about all the people who shows that quality, so "the poor" means "poor people", everybody who is poor.

TURN MY BACK= Literally it means to turn around so that you are facing to the opposite direction. But as an expression, if you turn your back when there is a problem, you don't want to know or do anything about it, you simply ignore it.

TO DIE TO MYSELF= In many religions, one of the objectives in life is "to die to yourself", that means that you stop living life self-centred (only thinking of you) and start living for the others, trying to make them happy and see to their needs. 

"For whoever wants to save his life will lose it, but whoever loses his life for my sake will save it" (Luke 9:24) 

"It is on account of the ego that one is not able to see God. In front of the door of God's mansion lies the stump of ego. One cannot enter the mansion without jumping over the stump." - Ramakrishna

"The common enemy of all religious disciplines is selfishness of mind. For it is just this which causes ignorance, anger and passion, which are at the root of all the troubles of the world." - The Dalai Lama