This statement is true.  This is one of the factors that cause the aggregate demand curve to slope downwards.  Economists often call this the “net exports effect.”  However, the statement that you give here does not reflect the complete nature of the net exports effect.

When prices in the US go up, American goods do, as you say, become more expensive relative to imports.  People start buying imports instead of domestic goods.  This causes the demand for domestic goods to decrease, which is reflected in a decrease in aggregate demand.  However, there is another aspect to this effect.  When American goods become more expensive, they also become less attractive to foreign buyers.  Foreigners do not want to import American goods if they are going to be expensive.  This means that an increase in the price level makes American exports decrease even as imports from other countries increase.  So there is something of a “double whammy” at work here.  The rising price level makes the demand for American goods fall both at home and abroad.  Therefore, the statement you have given is true, but is not a complete statement of the net exports effect.