Here's another interesting spin on the use of "Salamun alaikum". Brother Joe says that this phrase depicts a state of mind--it is NOT a greeting!
I noticed you spoke of this subject several times and I would like to share with you my thoughts on this.
The verses you shared (6:45, 7:46, 13:24, et al) do not speak of greetings at all brother. The words 'Salamun Alaykum' (peace onto you) simply refers to the mental frame of mind and/or spirit. When Abraham told his idol worshiping father 'Salamun Alaykum' - Abraham wasn't greeting his father but he was leaving him in a peaceful state of mind (19:47).
Similarly, God gives the Salam to his righteous servants (27:59) and his messengers (37:181). When Jesus said 'Salamun Alayah' (peace be upon myself) - he wasn't greeting himself, but he was declaring his peaceful state of mind and spirit (19:33). All this indicates that Salamun Alaykum are about peaceful frame of mind and spirit.
You also quoted 6:54 to support the idea that 'Salamun Alaykum' is a Quranic greeting. However, this is a wrong deduction for two reasons:
1. We should never omit any part of a passage. In 6:54 God doesn't tell us to say 'peace be upon you' and stops there. The verse tells us to say: "Peace be upon you, our Lord has decreed mercy upon Himself, that any of you who commits sin out of ignorance and then repents afterwards and reforms, then He is Forgiving, Compassionate."
2. 6:54 does not speak of a greeting but it's about the peaceful state of mind given to the believers and what we should all say when other believers come to hear the message of God:
6:54 If those who acknowledge Our signs come to you, then say, "Peace be upon you, our Lord has decreed mercy upon Himself, that any of you who commits sin out of ignorance and then repents afterwards and reforms, then He is Forgiving, Compassionate."
So as you can see, 'Salamun Alaykum' is NOT a greeting - at least not an exclusive one as you've concluded.
We should now ask ourselves if there is a true Quranic way to greet?! The answer is Yes!
The Arabic word for 'greeting' is TAHIYEH and this word has been used several times in the Quran.
In 24:61 we learn that a TAHIYEH (greeting) needs to be peaceful, blessed and good per God's standards... This kind of greeting is a greeting from God:
24:61 There is no blame upon the blind, nor is there any blame upon the crippled, nor is there any blame upon the ill, nor is there any blame upon yourselves, if you eat at your homes, or the homes of your fathers, or the homes of your mothers, or the homes of your brothers, or the homes of your sisters, or the homes of your uncles, or the homes of your aunts, or the homes of your mothers' brothers, or the homes of your mothers' sisters, or those to which you possess their keys, or that of your friends. You commit nothing wrong by eating together or as individuals. When you enter any home, you shall greet (SALLIMU) each other a greeting (TAHYIEH) from God that is blessed and good. God thus explains the revelations for you that you may reason.
So according to 24:61 a greeting in God's standards needs to be peaceful, blessed and good. This is why many Muslims say 'Al Salamu Alaykum Wa Rahmatul Allahi wa Baraktuhu' which translates to 'peace be upon you and God's Mercy and Blessings'... Such a greeting does NOT violate the Quran at all.
So a greeting can be 'Salamun Alaykum', 'Al Salamu Alaykum', 'Salam', 'Al Salamu Alaykum wa rahmatullahi wa baraktuhu'... These are all peaceful, blessed and good statements... I don't see any limit as to how the greeting is said so long as it is peaceful, blessed and good.
The Quran says...
That the dwellers of paradise will greet (TAHAYI) by only saying 'Salam' (14:23).
14:23 Admission was given to those who acknowledged and did good works to gardens with rivers flowing beneath them. In them they will abide by the leave of their Lord. Their greeting therein is "Peace."
Those who conspired against the prophet Muhammad NEVER greeted him a godly manner (peaceful, blessed, and good) (58:8).
58:8 Have you noted those who were forbidden from holding meetings in secret, but then insist on it? They meet secretly to commit sin, transgression, and disobedience of the messenger. When they come to you, they greet you with a greeting other than what God greets you with. They say inside themselves: "Why does God not punish us for our utterances?" Hell will be sufficient for them, wherein they burn; what a miserable destiny.
When someone greets us with a greeting, we shall return an equal or better greeting (4:86).
4:86 If you are greeted with a greeting, then return an even better greeting or return the same. God is Reckoning over all things.
All this information teaches us that although 'Salamun Alaykum' can be used as one of the greetings as per God's standards, it is NOT an exclusive greeting. God tells us that our greetings need to be peaceful, blessed and good and He doesn't specify how we must say them. 'Salam', 'Salamun Alaykum', 'Al Salam Alaykum', 'Al Salamu Alaykum Wa Rahmatullah', do not violate the Quran at all. """