How to use "doch"
I know "doch" is used to contradict a negative statement:
A: Das ist nicht wahr.
It's a great word for this usage and some languages really lack this word.
But, I found it hard to use "doch" in sentence, when it can be removed without hurting the sentence.
A: Wann kann ich mal zu Ihnen kommen?
B: Kommen Sie doch morgen um 10 Uhr.
Is it something like "anyway, come tomorrow morning at 10"? Is it necessary to use it there?
Doch, halt, eben, schon, mal, ja, nun - it's usage, usage, usage. Difficult to learn from books, much easier in spoken language where intonation and context will let you guess the meaning.
I often think of it as similar to the way we use the English word "do" - as in, "I do want to go to New York" rather than just "I want to go to New York."
There are already good answers to the question, but I would like to add a slightly more general one.
The word "doch" is in this context an example of a modal particle. These are words that are added to a sentence to convey mood or emphasis. They have no grammatical purpose so the sentence you get by removing the particle is always a proper sentence. But you lose the subtle nuances that modal particles convey. So the answer to your question is yes, you can remove "doch", but it will alter the way your sentence is received by the listener.
Examples of particles taken from canoo.net: doch, bloß, halt, mal, nicht, sehr, überaus, sogar, selbst, auch, erst, schon, überhaupt,
The meaning of these particles can be complex and highly dependent on context so many second language learners have problems understanding them properly (at least you and I seem to think so, Gigili). This excerpt from the Wikipedia article about German modal particles shows some of the complexity of the word "doch":
Doch can have several meanings. For one, it can be used affirmatively, or it can convey emphasis, urgency or impatience, or it can serve as a reply to a real or imagined, or pre-emptively answered, disagreement, hesitation, or wrong assumption on the part of the listener, or other people. In other situations this can have different effects.
Gehst Du nicht nach Hause? Doch, ich gehe gleich. ("Are you not going home?" "Oh, yes, I am going in a moment".) (Affirmation of a negative question; obligatory.)
Komm doch her! ("Do come here!") (Emphatically)
Komm doch endlich her! ("Do come on! Get a move on!") (More emphatically and impatiently)
Ich habe dir doch gesagt, dass es nicht so ist. ("I did tell you that it's not like that.")
Ich kenne mich in Berlin aus. Ich war doch letztes Jahr schon dort. ("I know my way around Berlin. I was here last year, after all/as a matter of fact.")
In other contexts, doch indicates that the action described in the sentence was, in fact, unlikely to occur:
Du bist also doch gekommen! ("You came after all.")
Ich sehe nicht viel fern, aber wenn etwas Gutes kommt, schalte ich doch ein. ("I don't watch much TV, but I do tune in if something good comes on.")