Yes and no.
The concept is known as Frame Dragging and has been theorized for a long time (since 1916).
A scientist came up with a way to measure this in 1956 but while the concept is simple enough the engineering necessary to build something sensitive enough to detect this took another 40 years. They had to manufacture the most perfect spheres (literally perfect not being possible) for the gyroscopes to pull this off. Each ping-pong ball sized sphere does not vary by more than 40 atomic layers from being a perfect sphere.
This was necessary to achieve the extraordinary precision necessary for the experiment. From the link just above:
Gravity Probe-B was launched in 2004 and it confirmed this frame dragging effect.
Important to note here though is the incredible precision necessary to measure such a tiny effect that is being caused by the entire earth.
The tilt-a-whirl may do the same thing I suppose the same as you experience time dilation when you walk to the bathroom but the effect is so insanely minor at this scale I think you can say that for all practical purposes the tilt-a-whirl is not curving spacetime.