The Space Race definitely played some role in the military rivalry between the east and west.
After the Soviet Union successfully launched Sputnik into space, Americans were worried that the Soviets would use this technology in order to launch an attack on America. In order to "keep up" with the Soviet Union, the US government investing millions of dollars in scientific and engineering research to develop the same technology as the Soviets. This constant competition between the US and Soviet Union allowed for the increase military capabilities for each country.
However, other events also affected this military rivalry. The US placing missiles in Turkey and the Soviet Union putting missiles in Cuba also caused a significant amount of tension between the two. This event almost lead to an all out nuclear war between these two countries. However, this crisis (known as the Cuban Missile Crisis) came to an end when President Kennedy and Khrushchev agreed to remove missiles from the aforementioned countries.
