Here are some thoughts; let me first tell you that they are all written rather critically - to illustrate possible criticism. Every and each of the arguments I present can be ignored (as faults) if you have a reason (that I failed to see).
Furthermore - in writing I consider that opening paragraph (sometimes even a sentence) and the closing paragraph of the whole work are the most important parts (first paragraph, I believe, is the most important).
This is where narrator will win the trust and interest of the audience and all further reading and interpretation of the reading will depend on success of this task (the trust does not need to be indisputable, but needs to be pragmatically established in some way).
Flowing more smoothly:
Main problem I find is that you are not consistent in your flow - you are somewhat switching the voice between writing it as stream of consciousness and classic retelling.
Let me illustrate:
The phone rang six times breaking the silence and darkness of the room for about ten or twelve seconds. I didn’t pick up.
Usually the text flows if sensory descriptions work together with thought descriptions in a natural way; however when you say that ”phone rang six times breaking the silence and darkness of the room for about ten or twelve seconds”, this is obviously a synthesized afterthought. Also, “ten or twelve seconds” is so precise that it implies importance of this fact, but actually it is not important. Even the simple thing like “phone rang six times” denotes that this is not stream of consciousness, but that narrator counted the rings and then retold it.
The phone broke the silence and darkness of the room. The phone rang again. And again, and again. I didn’t pick up.
This might not be what you are after, but refraining from mentioning 10-12 seconds and 6 rings, makes it more coherent to me (does the narrator really have to count how long it rings? the number of rings? does it matter for the story?)
Second paragraph, to make it flow better loose the parentheses - don't branch the flow in such a way unless that is exactly what you are after - parsing parentheses usually requires reader to switch context - from the mainline you are given some facts which are not pertinent to the mainline of the current thought. This is rarely justified in fiction.
Also, you give a bit too much detail which is not strictly coherent - ”As I walked on the brick floor, I looked at my left, and noticed that no food stand was open.”
How important is it really that narrator looked at the left, or that he walked on the brick floor? And again this would not be stream of consciousness - when I walk on the brick floor I don't think 'I am walking on the brick floor'; I might notice that it is a brick floor, by noticing something else about it. Ultimately I think you are just trying to describe the scene by throwing in the details. However, try to do it more naturally. Also, the type of floor does not matter much, but looking to left does; as later on narrator says: “I walked few meters, and turned my head to the other side” - this can not be understood (pictured) without remembering that he looked left before. But, none of these facts is important in themselves - the effort to understand the writing should come out of interest for the subject. I personally did not care (did not get intrigued) if the sea is to the right of the narrator's current (and unknown) orientation; I did care why did he walk a few meters before turning his head, but I that is not explained.
It was like a peaceful desperation. Like a silent scream. Like being asphyxiated in a huge empty space.
Let's synthesize that: peaceful silent desperate scream yelled (silently) while being asphyxiated in huge empty space. It is not easy to go through a single metaphor that employs near antonyms, let alone three of them in succession - it does not really work to make the thought any more precise or better defined. Try to be simple. For example, I personally liked the last the best and keeping only that one might be more powerful than listing many for the reader to choose from.
Here you mention again the ”six rings” and I must say that if you are going to refer to them again or if there is some symbolism in the number of rings, you should ignore my objection to counting them in the first paragraph. Still the question remains, is the number important?
Another stylistic issue is that universal quantifiers turn narrator less trustworthy - avoid phrases such as 'each time', 'never', 'always' and so on.
As usual, she didn’t even noticed that I had left the room. In fact, she had stopped noticing my presence in the apartment around a year ago. Each time I was sleeping beside her, I felt like there was a stranger lying beside me.
Now, when you say, "as usual" this implies that sometimes it is not as usual, but the opposite. So, according to that, sometimes, she notices. Then comes the sentence that she had stopped noticing him a year ago. Now it is not clear if it is a permanent state between the two or just usual state. Furthermore, do they talk? If yes, is not noticing the right phrase?
Finally - as I said in the introduction the critique I presented is highly imperfect; I could refute all of the given objections and/or justify them with further development of the text, so you can (and I encourage you to) ignore any of the above objections that you already considered; except, in my opinion, the importance of the first sentence/paragraph. Here, for example, anyone who gives the trust to the text will be thinking about the phone call; the whole opening is colored with it and that does not go away: ”Someone called me at half past eleven” is a good opening - it intrigues: who and why?