Sweden became the first major Western European country to recognise Palestine in October, but Prime Minister Stefan Loefven stressed that the recognition came with responsibilities.
"According to us Palestine is now a state. Our expectations of Palestine and their leadership will therefore increase," Loefven told reporters during Abbas' visit to Stockholm.
"There is no contradiction between keeping good relations with Palestine and keeping good relations with Israel," Loefven added.
Sweden's move to recognise Palestine last year prompted Israel to temporarily withdraw its Stockholm ambassador as relations between the two countries cooled.
Loefven announced a new aid programme to the Palestinians worth 1.5 billion kronor ($180 million, 159 million euros).
The deal stretches until 2019 and will go toward projects to fight corruption and promote gender equality and human rights.
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Loefven said both Palestine and Israel must be "ready to compromise" and that Sweden would support Palestine in "setting a regional example when it comes to women's rights."
Abbas, who arrived late Monday, thanked Sweden for the aid package and expressed the wish that "other countries would also recognise Palestine as a state."
"I am reaching out to Israel because we cannot achieve peace if we don't sit down for negotiations," Abbas added.
His visit comes a month after a senior Israeli official said Swedish Foreign Minister Margot Wallstroem was not welcome for official visits to the country.
Wallstroem last month postponed a trip to Israel amid Israeli press reports that her counterpart Avigdor Lieberman did not want to meet her.
Ahead of Abbas' visit, Israel's Stockholm Ambassador Isaac Bachman described the Palestinian leader's diplomatic efforts as a "diversion" from direct talks with Israel.
According to the Palestinian Authority, around 135 countries have recognised the state of Palestine including several that are now EU members.
Peace talks between Israel and the Palestinians collapsed in April last year, and a bloody war in Gaza erupted just a few months later.