Is there such a group? Or they never could be organized outside religious groups? Is there a moral argument, deprived of religious views, well accepted for anti-abortion?
It's worth remembering that historically abortion was substantially restricted (you could say "banned") in at least one Soviet-sphere country, i.e. Ceausescu's Romania. Ceausescu's "program" was apparently fairly effective at increasing the birth rate, in the short run. The Soviet Union itself had a somewhat milder incentive, namely the "tax on childlessness", also replicated (at least) in Poland in that era.
Probably more reliable than Wikipedia on Romania issue, https://srh.bmj.com/content/39/1/2.full
In October 1966, 1 year after coming to power, in an attempt to boost fertility, Romania's communist leader Nicolae Ceausescu made abortion broadly illegal, permitting the procedure legally only under a narrow range of circumstances: for women with four or more children, over the age of 45 years, in circumstances where the pregnancy was the result of rape or incest or threatened the life of the women, or in the case of congenital defect.
Just months after abortion was restricted, the number of safe, registered abortions had fallen 20-fold, and 1 year after the law took effect, the total fertility rate (TFR) nearly doubled (from 1.9 just before the restrictive law to 3.6 in 1967–1968). As women gradually found solutions for regulating their fertility – either through contraceptives procured illegally or through illegal abortions – the TFR began to fall again, reaching 2.9 in 1970, 2.2 in 1980–1984 and stabilised around 2.3 births per woman during the period 1985–1989
From a more ideology-oriented paper, the justifications of Ceausescu's law:
According to communist ideology, population has a triple role: work-force, a subject of income, and a consumer of the goods that were created, so the state ‘has to provide the policies for the population, in the same way it plans the national economy, in order to respond to the overall interests of society’ (Trebici, 1971: 39). The discussion was permanently centred on the way in which this intervention should be done. In the report of the Committee on the Study of the Measures designed to Improve the National Birth Rates, presented in the session of the Executive Committee of the Central Committee of the RCP of August 1966 (the basis for the preliminary discussion on the adoption of pronatalist policies), it was stated that the entire set of measures destined to raise birth rates was determined in an objective manner by the need to ensure an adequate work force for the development of the Romanian economy, especially after 1980. In the RCP manifesto of 1974, the sustained growth of the population was thought to be the essential factor for the dynamism and the productive strength of society. The target was the attainment of an active work-force of 11.5 million persons by the year 1990 (Partidul Comunist Român, 1975: 92).
The same paper mentions that apparently the Soviet Union had much more restrictive policies before the WWII, which apparently inspired Ceausescu.
The Soviet answer to the birth rate issue is exceptionally important for understanding the measures adopted by the Ceausescu regime. The Stalinist model of the 1930s-40s was a source of inspiration, from its severe restriction of abortions and the limitations it imposed on divorces, to its large-scale use of pronatalist propaganda (Goldman, 1993).
And the Soviet Union indeed had fairly restrictive law proposed in 1935:
The previously existing law “on the Legalization of Abortions” dated from November 1920. While referring to abortion as an “evil” and calling for propaganda to combat it, the law asserted that to protect the health of women abortions would be performed “freely and without any charge in Soviet hospitals.” By the mid-1930s, official concerns about a declining birth rate as well as the aim to strengthen the family unit as a bulwark of social stability doomed the old law. To Stalin, giving birth was “a great and honorable duty” which was “not a private affair but one of great social importance.” Henceforth, Soviet women would carry the double burden of holding a job in the wage-labor force and working in the home raising children. The draft decree proclaimed that “only under conditions of socialism, where … woman is an equal member of society … is it possible seriously to organize the struggle against abortions by prohibitive laws as well as by other means.” It permitted abortions only in cases when the continuation of pregnancy threatened the life of the pregnant woman.
The number of officially recorded abortions dropped sharply from 1.9 million in 1935 to 570,000 in 1937, but thereafter began to climb, reaching 755,000 in 1939. Despite criminal liability for performing illegal abortions, the actual number was probably a good deal higher.
That law adopted in 1936 and...
was colloquially referred to as ‘the law of happy motherhood’ [Zakon schastlivogo materinstva]
Apparently, this law persisted until Stalin's death, being repealed only in 1955.
Also, according to Wikipedia (citing the UN):
Laos has the strictest limitations on legal abortions among current Marxist–Leninist countries; it is only allowed to save the life of the mother.
So the 1930s Soviet tradition still seems to endure in some places.
Pretty much all of them. Being pro-abortion is a fringe position among liberals. Bill Clinton expressed a wish that abortion be "safe, legal, and rare". Most liberals strongly oppose abortion, and liberal groups engage in a myriad of anti-abortion activity, such as promoting sex education and birth control, while conservative groups help cause abortions by opposing those efforts.
But you're probably engaging in the conservative rhetorical subterfuge of using "anti-abortion" to refer to supporting the use of violence to prevent people from getting abortions. Since using violence to prevent bodily autonomy goes against liberal principles, any group that is pro-forced-pregnancy is by definition not fully liberal.
But we can still ask whether there are groups that are otherwise largely liberal. Since owning one's own body is one of the most basic rights, there aren't many groups that support liberty but oppose abortion rights. However, "liberal", in the American political dichotomy, is generally used to refer to both supporting personal liberty (socially liberal), and supporting governmental intervention in issues such as poverty (economically liberal).
Those two clusters of positions do not always go together. Libertarians are socially liberal but economically liberal. The opposite position of socially conservative but economically liberal doesn't have as well established of a term (there are terms such as "authoritarianism", but that is often interpreted as just being right-wing), and doesn't seem to be as common.
One demographic where the combination of socially conservative but economically liberal is more common is Catholics. Ethnic minorities are also more likely to be economically liberal than socially liberal; black and Hispanic people tend to be poorer than white people, and so be more prone to support positions that help the economically disadvantaged, but they also are often strongly religious, which is associated with social liberalism. One political party with this position is the American Solidarity Party, whose platform expresses concern about racism, the criminal justice system, war, voter suppression, and environmental issues. It also supports unions, immigration, and social safety nets
However, it also calls for laws against abortion, no fault divorce and various fertility services and declares pornography to be "inseparable from human trafficking, the promotion of pedophilia, and rape". It also includes code words promoting government endorsement of religion, prohibition of same sex marriage and medical care for trans people, conservative propaganda in sex education courses, censorship of views they consider to be anti-Christian, and the legalization of anti-LGBT discrimination.
Are there anti-abortion leftist groups?
In Wikipedia's List of political ideologies, there appears to be no leftist, nor left-leaning, ideologies for which opposition to abortion is a fundamental issue. However, there may be some variants in some counties that are both leftist and anti-abortion.
Or they never could be organized outside religious groups?
Politically, opposition to abortion is an "issue". Issue oriented sub-groups may arise within any ideology. Groups are comprised of individuals. Two leftist atheists could form a group opposed to abortion. They would not have much political influence.
Is there a moral argument, deprived of religious views, well accepted for anti-abortion?
There may be. However, this particular question is outside the realm of political consideration, thus off-topic for Politics SE.
In South America there is "Liberation Theology" which links a broadly left-wing or Marxist approach to social and political issues with Christian (in practice Catholic) theology.
I haven't been able to find much linking liberation theology with the issue of abortion; I suspect that leaders of the movement want to stay away from a potentially divisive issue. I did find this paper arguing that abortion should be legal as a practical reaction to the reality of patriarchal oppression (including sexual assault), and one or two other pieces by the same author along the same lines. However this seems to be something of an exception.
Pope Francis, despite coming from Argentina and having a broadly left-of-centre approach to political and economic issues, is opposed to Marxism and liberation theology generally. So there is a degree of tension between the Church and the liberation theology movement.
Of course! "Anti-abortion" and "Left wing" are not mutually exclusive.
A decent summary of what left wing actually is about:
Left-wing politics supports social equality and egalitarianism, often in opposition to social hierarchy. Left-wing politics typically involves a concern for those in society whom its adherents perceive as disadvantaged relative to others as well as a belief that there are unjustified inequalities that need to be reduced or abolished.
So all you need to do is find a group that believes in equality and egalitarianism of humanity yet is also opposed to abortion. Most anti-abortion religions have associated charities which support or at least advocate for the poor, and those charities should qualify.
After doing a bit of research; this is hard to answer for me locally, as "I oppose abortion in all cases" is as fringe a belief in Australia as "Lizard people control politics" is in the USA. Many of the religious charities I'm familiar with in Australia are actually conditionally supportive of abortion (or silent on it); however I was able to find a large organised group which opposes abortion and both directly supports the disadvantaged and wants to break down the income divide between rich and poor, The Salvation Army