In a fiat monetary system (for example, US or Australia) with an on-going external deficit, if you desire the domestic private sector to reduce its overall debt levels without employment losses, then you have to support the national government continually increasing the budget deficit in line with the private de-leveraging process.
Answer: Maybe
The answer is Maybe.
For the domestic private sector (households and firms) to reduce their overall levels of debt they have to net save overall. The behavioural implications of this accounting result would manifest as reduced consumption or investment, which, in turn, would reduce overall aggregate demand.
The normal inventory-cycle view of what happens next goes like this. Output and employment are functions of aggregate spending. Firms form expectations of future aggregate demand and produce accordingly. They are uncertain about the actual demand that will be realised as the output emerges from the production process.
The first signal firms get that household consumption is falling is in the unintended build-up of inventories. That signals to firms that they were overly optimistic about the level of demand in that particular period.
Once this realisation becomes consolidated, that is, firms generally realise they have over-produced, output starts to fall. Firms lay-off workers and the loss of income starts to multiply as those workers reduce their spending elsewhere.
At that point, the economy is heading for a recession.
So the only way to avoid these spiralling employment losses would be for an exogenous intervention to occur. While the question suggests that this intervention would come from an expanding public deficit, it could come from an expansion in net exports.
It is possible that at the same time that the households and firms are reducing their consumption in an attempt to lift the saving ratio, net exports boom. A net exports boom adds to aggregate demand (the spending injection via exports is greater than the spending leakage via imports).
So it is possible that the public budget balance could actually go towards surplus and the private domestic sector increase its saving ratio if net exports were strong enough.
The important point is that the three sectors add to demand in their own ways. Total GDP and employment are dependent on aggregate demand. Variations in aggregate demand thus cause variations in output (GDP), incomes and employment. But a variation in spending in one sector can be made up via offsetting changes in the other sectors.
So the best answer is maybe.
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