The Qur'an makes it clear that one may initiate greetings of peace with non-Muslims:
[Abraham] said, "Peace will be upon you. I will ask forgiveness for you of my Lord. Indeed, He is ever gracious to me." - 19:47
“And the servants of the Most Gracious are those who walk on the earth in humility , and when the ignorant address them, they say, “peace!” - 25:63
“And when they hear vain talk, they turn away from there and say: “To us our deeds, and to you yours; Peace be to you: we seek not the ignorant.” - 28:55
Yet we read in the hadith literature like in Saheeh Muslim, Book 26, Number 5389 the following:
Abu Huraira reported Allah's Messenger (may peace be upon him) as saying: Do not greet the Jews and the Christians before they greet you and when you meet any one of them on the roads force him to go to the narrowest part of it.
Does this not contradict what we have read in the Qur'an?
Before proceeding, one must bear in mind that there are a number of scholarly opinions regarding the rules of greeting a non-Muslim , however for the sake of argument we will proceed with the majority opinion which states that the Muslim must not greet the People of the Book with the greetings of peace before they do. This opinion does not contradict the aforementioned Qur'anic verses as I shall demonstrate insha'Allah.
Let's take a look at 19:47...
[Abraham] said, "Peace will be upon you. I will ask forgiveness for you of my Lord. Indeed, He is ever gracious to me."
Here "peace be upon you" or assalamu 'alayka isn't a salam of "greeting", but rather a salam of assurance and security that Ibrahim ('alayhi assalaam) would let his father be on his own and make up his own choice and that his father shouldn't expect any form of harm (whether in hurtful words or physical harm) coming from Ibrahim ('alayhi assalaam) toward him.  Another opinion was that this was a salam of farewell  or abandonment  and not a salam of greeting. Another possibility is that Ibrahim ('alayhi assalaam) prayed for peace upon his father as a way to soften his father's heart and attract him to his call , hence it was a salam of prayer and not a salam of greeting.
In conclusion, there is no strong reason for us to believe that Ibrahim ('alayhi assalaam) in 19:47 was necessarily giving his father a salam of greeting. With that in mind, there is no reason for us then to conclude that this passage necessarily stands at odds with the hadith stating that we shouldn't greet Christians and Jews first.
Let's now take a look at 25:63, which states:
“And the servants of the Most Gracious are those who walk on the earth in humility , and when the ignorant address them, they say, “peace!”
It must be borne in mind that the word "peace" or salaama does not necessarily have to be read as a quotation. For instance, look at the translation by Saheeh International:
And the servants of the Most Merciful are those who walk upon the earth easily, and when the ignorant address them [harshly], they say [words of] peace,
or the translation by Hilali-Khan:
And the slaves of the Most Beneficent (Allah) are those who walk on the earth in humility and sedateness, and when the foolish address them (with bad words) they reply back with mild words of gentleness.
This isn't a twisting of the Arabic by having a personal interpretation being imposed upon the text. Rather, the literal translation would be "say peace" and "say peace" doesn't have to be understood as meaning literally "say the word 'peace'", but could also mean "speak peacefully". 
Secondly, even we if assume for the sake of argument that it does literally mean that the word "peace" should be said, then the same scenario exists here as it does in 19:47. Who said that it's a "peace" of greeting? Linguistically, salaama could either be a salam of greeting or a salam of disavowal (i.e. that one is disassociating himself from those people and wishing them a permanent farewell).  So again, we ask why it should be taken as a salam of greeting.
Even with Surah 28:55, the scenario is the same as the previously two mentioned verses. 
In conclusion, no convincing case has been put forward demonstrating a necessary contradiction between the Qur'an and the ahaadith on the subject of initiating greetings to non-Muslims.
1) See my article... Rules of Greeting Non Muslims In Islam.
2) See the commentaries of Abu Bakr al-Jazaa'iri, Ibn Kathir, al-Baghawi, al-Baidaawi, as-Sa'di & az-Zamakhshari on 19:47.
3) Abu Hayyan al-Andalusi in his commentary al-Bahr al-Muheet, on 19:47 said: وقيل : هي تحية مفارق
4) Imam al-Baghawi in his commentary Ma'aalim at-Tanzeel, on 19:47 said: وقيل : هذا سلام هجران ومفارقة
5) Az-Zamakhshari in his commentary al-Kashaaf, on 19:47 said: ويجوز أن يكون قد دعا له بالسلامة استمالة له.
6) This opinion was adopted by several scholars.
7) Imam al-Qurtubi in his commentary al-Jaami' li-Ahkaam al-Qur'an, on 25:63 said: قال النحاس : ليس " سلاما " من التسليم إنما هو من التسلم ; تقول العرب : سلاما , أي تسلما منك , أي براءة منك
8) See the commentaries of as-Sa'di, al-Baghawi and Ibn Ashur on 28:55.