The government's measure of new home sales fell in April to the lowest level since it began keeping such figures 19 years ago, giving further evidence of the housing industry's depression.

And in another bad sign for the nation's economy, the Commerce Department reported that new factory orders in April fell 2.3 percent from March.

The factory orders report was particularly discouraging for the economy because the government's measure of new orders had risen in March and February.

One of the most depressed job markets is home construction, which is reeling from the effects of a weak housing industry and real estate market.

The Departments of Commerce and of Housing and Urban Development said in their report on home sales that the decline in April sales of new single-family homes was 15.3 percent from March, to an annual rate of 315,000.

That rate, which is adjusted for seasonal variations, is far below the previous record low of 335,000 last September. The March rate was 372,000.

Robert Ortner, the Commerce Department's chief economist, held out little hope for a quick turnaround for housing, saying "prospects for an upturn in the immediate future are virtually nil."

Jack Carlson, chief economist at the National Association of Realtors, suggested that President Reagan cancel his European trip "and concentrate fully on his failure and the failure of Congress to adopt a 1983 budget that would lower deficits, government borrowing and interest rates."

Reagan left yesterday for a weekend economic meeting of Western leaders in Europe.