After examining all of the Scriptures which mentioned the Amalekites, I came to the conclusion that the one thing which is common to all of these references is that the Amalekites were consistently against Israel. Whether alone or allied with other peoples, there is never a time when their attitude toward Israel was not hostile. For that reason, I figured that the reason that the LORD was hostile toward them was because He was keeping His covenant with Abraham and with Israel, which included the promise that "I will bless those who bless you and whoever curses you I will curse.." (Genesis 12:1-3) I was still a bit unsatisfied, though, with this reason. I mean, it was totally valid, yet it did not seem to be the whole picture.
Then I remembered another book of the Bible which deals with this enmity between Israel and the Amalekites. It is the book of Esther. I knew that this book did not refer to the Amalekites by name, yet someone who is an Amalekite figures prominently in the story.
In Esther, a person named Haman is mentioned. He is described as "son of Hammedatha, the Agagite". While Hammedatha is probably Haman's father, the 'Agagite' part of his description most likely refers to Agag, who was king of the Amalekites back during Saul's time as king over Israel. (I Samuel 15:20) Saul had killed most of Agag's people, the Amalekites, yet had spared King Agag, despite the LORD's orders to kill all of the Amalekites. The prophet Samuel had killed Agag in order to complete what the LORD had commanded, and Saul's disregard for the LORD's commands became the main reason for his own loss of kingship over Israel.
Haman is promoted to a position higher than that of the other nobles in Persia. They would all bow down and pay him honor when he arrived, as king Xerxes had commanded. However, Mordecai, a Jew from the tribe of Benjamin whose ancestors had been taken captive from Jerusalem by Nebuchadnezzar king of Babylon, refused to bow down to him. This enraged Haman, who wanted to destroy not only Mordecai, but his whole people. (Esther 1:1-6) In fact, verse 6 notes that "...having learned who Mordecai's people were, he scorned the idea of killing only Mordecai. Instead Haman looked for a way to destroy all Mordecai's people, the Jews, throughout the whole kingdom of Xerxes." This, too, is further evidence that Haman is an Amalekite, and likely a relative to King Agag. Otherwise, why would Haman care at all who Mordecai's people were? (His likely relationship to King Agag also is strengthened by the fact that Haman seems to be of noble birth himself, for he is counted among the nobles.) (Esther 1:6)
Mordecai's actions also seem to indicate that he considers Haman an Amalekite. The Jews did not consider it wrong to bow down to show respect for a king or other person (unless it violated the 2nd commandment about not bowing down to other 'gods'). Yet Mordecai absolutely refused to bow down to Haman. The NIV Study Bible's comments on these verses puts the situation into words better than I can:
"Only the long-standing enmity between Israel and Amalek accounts both for Mordecai's refusal and for Haman's intent to destroy all the Jews. The threat against the Jews 'throughout the whole kingdom' (v.6) is a threat against the ultimate issue of redemptive history."
Then I really had an 'aha' moment, for I realized that this 'threat against the ultimate issue of redemptive history' was really the crux of the matter regarding the LORD's wrath against the Amalekites. How? Well, if the Jews were wiped out, as the Amalekites (and Haman) wish, then The LORD's redemptive purposes would have been thwarted. The Messiah, the Savior, the ultimate 'Anointed One', was to come through the Jewish people. Therefore, if any person or people group stand against the LORD's purposes in this matter, it is understandable that they would become the object of His wrath.
Here is what I would say in summary: The Amalekites were no greater sinners than any other people group. The Scriptures say that "All have sinned and fall short of the glory of God". (Romans 3:23) Also, if any individual among them were to turn to the LORD, they could be forgiven by means of the same sacrifice which makes any other person righteous, the one made by the Messiah, the Anointed One, Jesus:
has been made known,
to all who believe.
through the redemption that came by Christ Jesus.
he had left the sins committed beforehand unpunished --
No, but on that of faith.
and the uncircumcised through that same faith."
We can see by these Scriptures that all people are welcome in God's kingdom. In those Scriptures about the Amalekites, they were standing in the way of God's redemptive plan and He would not tolerate that. Yes, God was also keeping His covenant with Israel, blessing those who blessed them and cursing those who cursed them, but behind that was the even more far-reaching matter of unfolding His plan of redemption for all peoples. That is why He reacted with such vehemence to threats against the Jewish people.
Now, Jesus has already come. He has made that sacrifice on the cross to atone for the sins of all mankind. Jesus is God's invitation to every individual, regardless of race or religion, to receive forgiveness and be reconciled to God.
One further thought: The Jews are still God's covenant people. He continues to this day to uphold the promises which He made to Abraham and to Israel. That is why Israel continues to exist despite numerous and continued attempts to wipe them out as a people. That is also why they will continue to exist until the LORD returns, even amidst clamoring boasts and threats to the contrary. The God of Israel remains unchanged as well. It would be wise for all who seek to do harm to the people of God to learn well and pay heed to these warnings in Scripture.