No. Space-time is astronomical mathematical model or mathematical concept of the universe as the whole, not confined to Earth.
While the Earth and our solar system are parts of the universe, nothing we do on Earth affect our Milky Way in any measurable way.
Right now, we can send (unmanned) space crafts to any part of the solar system, but they don't in any way affect any of the planets. Even if all life on Earth died out for some reason, it would not affect the Sun, or our nearest neighbors, Mars and Venus. But if the Sun die, eg become a Red Giant (more unlikely scenario, go supernova or black hole), after running out of hydrogen atoms to fuse, the Sun will become dimensionally larger in size, and hotter than is now, it will kill all life on the planet. If the Earth don't swallow up by the red giant sun, the Earth will become like the new Venus, a lifeless, fiery hell. Becoming the red giant, the sun will affect all the planets in our solar system.
Now imagine the Earth, comparing it with the Milky Way. Do you think the death of all life on Earth will affect the galaxy?
I don't think so.
While space travel is all great and fascinating, we are no where near the technology of going to new stars. Colonising other planets from other stars, are currently no more than science fiction.
But going back to your question, atanu.
The Earth is not outside of space-time. Just that the theory of evolution has no affect to space-time, and space-time does not affect evolution.