I remember back in undergraduate school that GDP=C+I+G. So GDP may well fall if G falls. But I suspect every 1% fall in G may cause a substantial increase in C and I. Fewer bureaucrats with full time jobs devoted to messing up the real economy.
So even if the G component is falling, the real economy may be growing.
In what we would consider to be a normal world, you would be correct. Lower government spending would free up capital for investment that would grow the private side of the economy. However, we are in a new normal. Government spending and regulation have become so entwined with the private sector that the C + I depend on the government for a big part of their existence. Also consider that any newly available capital may likely be used to expand production overseas so as to maximize profit. In my opinion, domestic private sector growth is now tied to Uncle Sam. We are so screwed thanks to Krugman and the unintended consequences of economic and foreign policy over the last 20 years.