Here's what he said.
"Then we stand thirty-four to thirty-four, and your Black Republican Vice President to give the casting vote. Mr. Lincoln can make his own nominations with perfect security that they will be confirmed by this body, even if every slaveholding State should remain in the Union, which, thank God, they will not do. You have elected your President, and you can inaugurate him; and we will have neither lot nor parcel in this matter.
Notice he didn't say anything about the "High Tariff" Republicans. He didn't say that Lincoln was going to do anything to his state or had even considered doing anything to his state. He simply complains that "Slave States" were losing their ability to control the federal government. Now tell the class again how the Civil War was not about slavery.
That part was elsewhere in the speech:
"But if, in the mean time, you can bring the power of the Federal Government to coercion, and before the Treasury is drained of its last dollar, you can make soldiers out of your operatives and your sailors, you expect then, amid the heat of the contest, the confusion of ideas, as well as everything else, that you can conceal the facts, and denounce us for the calamities that are on this people; and you expect not to lead, but to send them to battle...Their own citizens have sworn faith and allegiance to the State, and obedience to the Constitution of the United States; but the oath is allegiance and fealty to the State of South Carolina. Now, you tell those men that, if they obey their oath, if they violate their allegiance, if they become not traitors to the country that has nourished and nursed them, then they are to be shot down like dogs or hanged like felons. This is the manner in which their remonstrances are met. This is the language of Senators who talk about ?our glorious Union,? and our being a race of freemen. This is the language used to free Americans! This is in a land of liberty! This is among a people who boast that they have the right of self-government! Well, you will have an opportunity of making the experiment."
He simply seems to not care for the results of an election.
No. He's outlining how the northerners controlled all of congress plus the presidency, allowing them to do whatever they wanted to the south. You and many others have denied this to have been so, yet it is a simple fact of history.