Transmission through inanimate objects is referred to as "fomite" transmission. It's really hard to know exactly how important fomites are for a particular illness. There is, however, strong evidence that respiratory viruses can spread via this method. Generally, the emphasis is on how contact with hands can lead to infection when someone touches their hand to their mouth, nose, or eyes.
The review by Boone & Gerba, 2007 touches on several issues of fomite transmission. In general, it is easiest to demonstrate evidence of the potential rather than the actual rate of fomite transmission, by looking at viability of viruses on surfaces and how well they transfer to other objects and the hands. However, they also write about epidemiological evidence:
Several epidemiological studies have supported laboratory studies by indicating environmental contamination as a potential vehicle for virus transmission. During an outbreak in a Honolulu nursing home, it was determined that staff hands or fomites (e.g., towels, medical cart items, etc.) spread influenza virus (51). An outbreak of coronavirus (SARS) in a Hong Kong apartment complex may have resulted from fecal-oral transmission combined with environmental contamination (62).
When natural rhinovirus colds were studied, rhinovirus was found on 39% of symptomatic individuals' hands (35). Additionally, volunteers touching contaminated objects and/or the fingers of symptomatic individuals had a higher attack rate of colds if they inoculated their own eyes or nose (35).
They also refer to studies of disinfection/hygiene intervention. The only practical way for these methods to interfere with transmission is by preventing fomite transmission, so if cleaning surfaces and washing hands reduces infections, you can infer that these are contributing routes of infection:
In addition, a review article by Barker et al. cited over 15 research studies that indicated a decrease in viral contamination and viral infection when hand washing was used regularly as an intervention (7). Subsequently, disinfection and hygiene intervention studies, which have cited a reduction in nonspecific illnesses, only support interruption of disease transmission.
I've also attached the reference from that Barker et al paper, which is a simple, informative read.
Barker, J., Stevens, D., & Bloomfield, S. F. (2001). Spread and prevention of some common viral infections in community facilities and domestic homes. Journal of Applied Microbiology, 91(1), 7-21.
Boone, S. A., & Gerba, C. P. (2007). Significance of fomites in the spread of respiratory and enteric viral disease. Applied and environmental microbiology, 73(6), 1687-1696.