Victor Ponta's leftist Social Liberal Union (USL) won two-thirds of the seats in Sunday's vote but is locked in a power struggle with Mr Basescu, a former sea captain who said he would never again appoint the 40-year-old lawyer.
Any impasse in forming the next government could leave Romania, clawing its way out of a deep recession, rudderless for weeks and delay talks for a new International Monetary Fund bailout once a €5 billion (£4bn) deal expires next year.
The USL tried to impeach conservative Mr Basecu in July, accusing him of overstepping his powers.
The standoff has raised questions over policy and the leu currency is only about 2% off its record low against the euro, while borrowing costs have edged higher in the past month.
"I expect President Basescu to respect the constitution and the will of the Romanians," Mr Ponta said. Under Romanian law, Mr Ponta's current government remains in power until the new cabinet is approved by parliament.