Sainid's answer mentions the common story that an Indian monk introduced exercises to the Shaolin temple, and their martial arts practice grew out of those exercises; this is probably what the Ukrainian guy is referring to. There are hundreds of web sites and books sharing that notion, for example (Bodhidharma) "taught various exercises to the monks there that would develop into modern day kung fu".
Wikipedia's article on Shaolin Kung Fu also mentions this popular myth of "Bodhidharma influencing Shaolin boxing" but largely discredits it. See the Wikipedia article for links to further reference material:
Some popular stories consider Bodhidharma as the founder of Shaolin kung fu.
The idea of Bodhidharma influencing Shaolin boxing is based on a qigong manual written during the 17th century. This is when a Taoist with the pen name "Purple Coagulation Man of the Way" wrote the Sinews Changing Classic in 1624, but claimed to have discovered it. The first of two prefaces of the manual traces this qigong style's succession from Bodhidharman to the Chinese general Li Jing via "a chain of Buddhist saints and martial heroes."15 The work itself is full of anachronistic mistakes and even includes a popular character from Chinese fiction, the "Qiuran Ke" ("Bushy Bearded Hero)" (虬髯客), as a lineage master. Literati as far back as the Qing Dynasty have taken note of these mistakes. The scholar Ling Tinkang (1757–1809) described the author as an 'ignorant village master'."(p168)
Like other stories of Shaolin, this story has, after all, some basis in reality. Bodhidharma was the founder of Dhyana (Chinese: 禅; pinyin: chán; Japanese: zen) Buddhism.
In the Wikipedia article on Bodhidharma:
According to Chinese legend, he also began the physical training of the monks of Shaolin Monastery that led to the creation of Shaolin Kung Fu.
Some Chinese myths and legends describe Bodhidharma as being disturbed by the poor physical shape of the Shaolin monks, after which he instructed them in techniques to maintain their physical condition as well as teaching meditation. He is said to have taught a series of external exercises called the Eighteen Arhat Hands and an internal practice called the Sinew Metamorphosis Classic. In addition, after his departure from the temple, two manuscripts by Bodhidharma were said to be discovered inside the temple: the Yijin Jing and the Xisui Jing. Copies and translations of the Yijin Jing survive to the modern day. The Xisui Jing has been lost.
Your question continues:
...it does seem a no-brainer that other societies across the world would have developed their own fighting systems before, or at the same time, as India. // ...there were fighting systems in place much before the incorporation of yoga or ayurveda into the fighting styles.
For sure, your Ukrainian friend wouldn't be totally mistaken to think that Indian practices like yoga show an extreme mastery of mind and body that could complement martial arts training, and seems quite rare internationally and historically. In my opinion, India is not an unreasonable place for a martial artist to seek knowledge. That said, I've not personally seen evidence of any especially developed martial arts in India, compared to other locations, but for some reason they seem to get less media attention from the Western world that many other countries' arts.