* Von der Leyen backs climate goals, single market reforms
* EU parliament to vote on new Commission head next week
* Future policies, decision-making at stake for 500 million people (Updates with Greens rejection)
By Daphne Psaledakis and Peter Maushagen
BRUSSELS, July 10 (Reuters) - Germany's Ursula von der Leyen made her pitch to European Union lawmakers on Wednesday to become the next head of the bloc's executive, but her promises to fight climate change were not enough for Green deputies who vowed to vote against her.
The conservative German defence minister may still win support of the full parliament next week, but rejection by the Greens leaves her relying more on nationalists in eastern Europe who like her tough stance on Russia.
Von der Leyen was "evasive on all the answers ... We don't know her," Bas Eickhout, a Green lawmaker from the Netherlands, told Reuters. "What she will do concretely is unclear."
The powerful European Commission presidency, which overseas trade negotiations, antitrust rulings and broad policy for 500 million Europeans, was offered to von der Leyen on July 2 by EU leaders. She would be the first woman to hold the post.
However, a fragmented European Parliament following the May election means she has a harder task in being nominated so losing the Greens, who have 74 of the 751 seats, makes support from the centre-left crucial.
They have yet to decide whether to vote for von der Leyen, who hails from the centre-right wing of European politics led by German Chancellor Angela Merkel and who would replace Luxembourg's Jean-Claude Juncker.
Von der Leyen needs the backing of an absolute majority of 376 votes in the 751-strong chamber to be confirmed.
The EU, weakened by the 2009-2012 euro zone crisis, Britain's decision to quit the bloc and the rise of far-right and far-left eurosceptic parties, needs a strong Commission president to improve its fortunes, officials and experts say.
In her first public policy comments since being nominated, von der Leyen said the EU needed to be united, as it contends on the global stage with a rising China, an unpredictable U.S. administration, and an assertive Russia.
"The EU is based on principles. This is the foundation - respect for the rule of law," she told a group of liberal deputies, adding that it was also vital to boost the competitiveness of the EU's economy.
CARBON NEUTRAL SUPPORT
Despite Green scepticism, she said she wholeheartedly supported the EU going carbon neutral by 2050 and said the bloc should harvest benefits of environmentally-friendly policies.
She also said the EU needed to advance single market reforms, invest in joint defence capacities that would complement those of NATO, and "become more assertive" towards the United States.
Von der Leyen, a 60-year-old former gynaecologist and mother of seven, switched between English, French and German as she spoke in favour of enlarging the euro zone and the EU's open-border Schengen area, provided countries meet the criteria.
She said the EU should also be ready to take in Western Balkan countries.
Von der Leyen promised to be tougher on the rule of law than the outgoing Commission and to reform the failed asylum system - both areas where liberal, wealthy western Europe and the post-communist east differ bitterly.
On Brexit, she hoped Britain would remain in the EU but, otherwise, she said it was essential the divorce did not poison the chances for good future cooperation between the bloc and London. "We want you to stay," she told one British lawmaker in the assembly.
She said the Irish border "backstop" - which aims to avoid border controls between EU member Ireland and British-governed Northern Ireland after Brexit - was "precious, important and has to be defended" in any Brexit deal.
So far, von der Leyen can count on the conservative European People's Party (EPP), who have 182 seats, and most likely the liberals, who have 108 lawmakers. But she needs more votes.
Failing to secure enough support would force an embarrassing delay or even push EU states to look for another candidate.
Should she succeed, von der Leyen said she wanted a gender balance at the Commission and that Dutch Socialist Frans Timmermans would be her first deputy along with Danish liberal Margrethe Vestager, now the EU's competition chief. (Additional reporting by Gabriela Baczynska, Robin Emmott, Alissa de Carbonnel, Alexandra Regida; Writing by Gabriela Baczynska and Robin Emmott; Editing by Gareth Jones and Andrew Cawthorne)