I watched his speech last night, and don't recall him saying anything I took as offensive to Great Britain, but I was also fiddling with my guitar so I thought I might have missed something. I guess this is the part of the speech you are referencing.
"America is more than just a land mass between two oceans, America is an ideal. A simple, yet powerful ideal. Freedom matters. For much of human history government power has been the unavoidable constant in life. Government decrees and the people obey, but not here. We have no king or queen, we have no dictator, we the people constrain government. Our nation is exceptional because it was built on the five most beautiful and powerful words in the English language, “I want to be free.” ".
I am not that big of a fan of Ted Cruz in general, so I have no reason to defend him, but I think you are misinterpreting what he was saying. I took it to mean pretty much that here in the US, government serves and works for the citizens, not the other way around. He wasn't referencing Britain specifically with his use of "King and Queen", just as he wasn't singling out any country with the term dictator, he was using those terms broadly to describe any government that gives power to the leaders at the expense of the people. This makes the argument for a smaller, not larger government. The only time he really referenced Britain, from the transcript I just skimmed over, was in referencing the freedom they gained with Brexit. Besides, the royals of Britain may have a small degree of influence, but as far as I know, their power today is only symbolic.