How we cite our quotes: (Paragraph.Line)
Quote #4
We have been, and propose to be, more and more American. We believe that we can best serve our own country and most successfully discharge our obligations to humanity by continuing to be openly and candidly, intensely and scrupulously, American. If we have any heritage, it has been that (4.1-4).
Openly and candidly American—as opposed to traveling in Europe with a maple leaf decal on your backpack? Coolidge isn't super specific about what it means to be "American," but still, Americans should still be as American as they can be. Given the context of the speech, he means that to be "American" means to defend liberty and American ideals around the world and to make sure every country has access to reasonable-priced Quarter Pounders—or Royales with Cheese, as the case may be.
A need for a shared culture of American-ness was behind Coolidge's immigration policies, too. He wanted to make sure that we only allowed immigration to the extent that these folks could be fully acculturated into American society.
Quote #5
Here stands our country, an example of tranquility at home, a patron of tranquility abroad (25.4).
True, the war was over and the nation was at peace, but there was also a lot of organized crime and violence happening within the country's borders. Not that Coolidge would want to highlight that in his speech, though. Most presidents leave that stuff out of their inaugural addresses. Most.