The Quran has abolished slavery!
The Quran has announced its target to solve the slavery problem at the beginning: “Faqqu raqaba: abolish slavery, abolish oppression, liberate everybody.” (al-Balad  13).
The main reason to get slaves is wars. The verse al-Anfal  67 has eradicated slavery: “It is not for a Nabi (prophet) to take prisoners before he has subdued the land…” If a Nabi hasn’t the right to take prisoners, as the applier of the rules in the war, then, of course normal Muslims don’t have the right either. The enemy could captivate only if he breaks down during the war, but not before or later. It was submitted for the prisoners only two options to be freed of captivity: Release by ransom (fidyah), or grace (Muhammad  4). There is definitely no permission to turn the prisoners into slaves. Rasulullah didn’t turn anyone, including the prisoners from the first war thus Badr, into slaves. However, the madhabs had been given theopportunity to continue slavery!
It was the target of Islam to liberate existing slaves according to a period of time with various matters, but not instantly:
* One of the ways to go to Paradise is to liberate and help slaves without discriminating against them whether Muslim or not. This is a method of worship (ibadah) to Allah (al-Baqara  177; al-Balad  13; an-Nisa  36).
* To make a freedom treatment (Muqataba) with them, who recognizes good (an-Nur  33). For example, some of the prisoners from the Badr war, were liberated with a reward for teaching 10 Muslim children to read.
* One of the 8 groups are slaves which could be given Zakat and alms, who want to be liberated (at-Tawba  60). The Muslims should be helpful and do favours to them (an-Nahl  71; an-Nisa  36; an-Nur  33).
* The slaves should be liberated as an atonement (Kaffarah) for some crimes (al-Maida  89; al-Mujadila  3; an-Nisa  92).
* The slaves could be liberated by marriage, but only on the condition of their free allowance (al-Baqara  221; an-Nisa  3, 25; an-Nur  32). It’s a social duty to help slaves about getting married (an-Nur  32).
If the existing slaves were to be liberated instantly, then thousands of them could be in miserable situations. For example, hundreds of thousands of slaves were liberated instantly after the American civil war (1861-1865) and became miserable, jobless, homeless and without food. In addition, liberated masses of slaves could be the cause of disorder and revolts.
During the period of Jahiliya a free person wasn’t executed, if he killed a slave or a woman. It was possible that an owned slave was killed instead of the murder of a high status and strong family or clan. In al-Baqara  178 and related verses (al-Isra  33; al-Maida  32) it can be seen that an intentional kill has to be retaliated (Qisas) without considering murderers identity and status.
The use of slavery in an-Nahl  75 and az-Zumar  29 is not for approving slavery, they are only for simile. The Quran uses a lot of similes (for example blind and non-blind person: al-Anam  50; Fatir  19-22).
Therefore, the Quran has definitely abolished slavery. It established rules to liberate the existing slaves in various matters.