May I advise you something? When learning a foreign language, there is no need to ask "why" everytime. All what you need is to ask "how".

Every language has its own rules and its own logic.

In English: "I am looking at you". Why to use "at"? Why not "on"? For me, it seems more logical to say "I am looking on you" (Just kidding, I know I have to use "at", but in Russian we would use "on").
Why do you say "John helps Mary" and not "John helps to Mary"? Again, the second version would be more logical for Russians.
Why do you "wait for a bus"? And not just "wait a bus"? (As we would say).
Why do you "answer a question" and not "answer on a question"? We do "answer on questions" in Russian.
Why do you "listen to music"? Why to use "to"? We don’t use it, we just "listen music".
Your things "depend on something", ours "depend from something".
Your things "consist of some components", ours "consist from some components".

All the examples above show the thing which is called "verb government". Every language has its own unique verb government. You just have to LEARN which verb requires which case and/or preposition.
"to look at smth." = "смотреть на + acc."
"to watch smth." = "наблюдать за + instr." Or "следить за + instr."

Don’t learn a verb alone! You cannot guess a correct government, you should learn a verb with its government. 2-3 sample sentences for each verb.

You would unevitably face the same problem when learning ANY other language (not only Russian). Even closely related languages often have different government patterns of some verbs.

Some links for you for further reading (if you are interested in this):
http://en.wikipedia.org/wiki/Government_(linguistics)
Case government - Wikipedia, the free encyclopedia

Quote Originally Posted by tiudavidharris View Post
do we use accusative case on 'на тебя' because of direction and because 'тебя' is the object of the sentence?
"тебя" in "Я смотрю на тебя" is not an object (at least, it is not a direct object, because there is a preposition).

But yes, "на тебя" shows some direction (i.e. the direction of where you are looking). And it is a common way to indicate directions using "на + acc." or "в + acc.": подниматься на гору, положить на стол etc.

It's like as if you would say "I am looking ONTO you". But you have just to learn it.

Better take 2-3 sentences and try to memorize, so you'll get a pattern:
Я смотрю на тебя. I'm looking at you.
Я смотрю на реку. I'm looking at a river.
Я смотрю на дом. I'm looking at a house.
Я смотрю на рисунок. I'm looking at a picture.