Not completely, you just haven't thought it all the way through. For starters, the earth is weightless, same as the astronauts floating about the orbiting space shuttle are weightless. The earth's mass, however, is about 6 x 10^24 kilograms. Suppose we launched the equivalent mass of an average sort of car, say 1000 kilograms, into interstellar space every day for 10,000 years. That'd be around 3,652,500 cars, or 3,652,500,000 kilograms. So, round it to 4 billion kilograms the earth would lose, just to make it easier, we're not worried about bang-on accuracy here. What fraction of the earth's mass is that? A simple division will tell you it's around one part in 10^15, one thousand trillionth. To put that in perspective, there are about 10 trillion cells in the human body, so it's a loss of mass on the same scale as you losing a hundredth of a single cell.
Like Tonington says, I wouldn't worry about it.