As with much research in economics, the evidence is messy and mixed. And so my answer will have to be an unsatisfying "we don't know".
In a review paper, Jones (2008) arrives at a similar conclusion:
The results from studies that evaluate the influence of the ADA in the US, are mixed. Early work suggests the legislation had a negative effect on employment but this is not always supported in more recent analysis, particularly using state level data. The evidence also highlights issues that are important in this type of evaluation including defining the disabled population on the basis of policy and controlling for changes in the composition of the disabled as a result of the legislation. To separate the effects of legalisation from more general trends over time pre-existing trends and business cycle effects need to be controlled for, as do the effects of other policy changes during the same period.
Highlights from a number of papers that have attempted to measure the effect of the ADA on employment, in chronological order:
On average over the post-ADA period, employment of men with disabilities
was 7.2 percentage points lower than before the act was passed.
Acemoğlu and Angrist (2001):
In 1993, the year after the ADA came into effect, there were marked drops in the employment of disabled men aged 21–39, both in absolute terms and relative to the nondisabled. A similar drop is observed in 1992 for disabled women aged 21–39. Extrapolating employment trends, allowing for composition effects, and controlling for changes in disability insurance and SSI participation rates do not seem to account for these declines, leaving the ADA as a likely cause. This interpretation is also supported by evidence that employment of disabled men fell more sharply in states with more ADA-related charge activity and by relative declines in the employment of disabled workers in medium-size firms.
Kruse & Schur (2003) find:
the main conclusion is that there is reason to be cautious about findings of either positive or negative effects given the limitations of existing measures in reflecting who is covered by the ADA.
Beegle & Stock (2003):
We find that disability discrimination laws are negatively associated with the labor force participation rates of the disabled, relative to other disabled individuals in states without such laws, although the impact is small. We also find negative effects of the laws on disabled relative earnings. However, unlike some previous research, we do not find a systematic negative relationship between disability discrimination laws and the relative employment rates of the disabled once we control for differential time trends in disabled and nondisabled employment. One exception is among females, where the estimates indicate a small negative effect of the laws on relative disabled employment rates. The effects of the laws on labor force participation and earnings are larger for whites than for nonwhites and larger for females than for males. Finally, reasonable accommodation clauses are not associated with lower relative disabled employment rates for either the full sample or any of the subsamples.
Houtenville & Burkhauser (2004):
The relative employment of working-age people with disabilities declined in the 1990s. Based on our review of the evidence, however, the ADA is not the likely cause of this decline. Instead, we find that the relative employment of the population with longer-term disabilities— a population that is more likely to be eligible for SSDI/SSI benefits and ADA protection—began to fall around the mid-1980s, well before the implementation of the ADA, but soon after 1984 legislation that substantially expanded the medical definition of disability used by the SSDI and SSI programs.
Hotchkiss (2004) abstract:
This paper replicates recent findings that employment among disabled people has declined since the ADA. A closer look indicates that this decline results from a drop in the labor force participation rate among those classified as disabled. Further analysis indicates that this labor force participation rate decline, however, was not the result of disabled individuals fleeing the labor market, but, rather; more likely a result of the reclassification of nondisabled, nonparticipants, as disabled. The unconditional employment probability among disabled people (taking selection into the labor market into account) has not declined, and may have actually improved for certain disability classifications.
Jolls & Prescott (2004):
the pattern of the ADA's effects across states suggests, contrary to widely discussed prior findings based on national-level data, that declining disabled employment after the immediate post-ADA period reflects other factors rather than the ADA itself. ... Our results suggest that while the ADA’s reasonable accommodations requirement had a significant negative effect on disabled employment in the near-term after the ADA’s enactment, the law may well have had no causal link to the declines in disabled employment through much of the 1990s.
Moon & Shin (2006):
Compared to nondisabled counterparts, employment for disabled men fell significantly by 4.8 or 5.8% with self-reported or objective measure, respectively. Most of adverse effects of the ADA on employment of disabled men are found to concentrate on full-time job employment rather than part-time job (i.e., 5.8% fall in work/no work employment versus 7.1% fall in fulltime job employment).
Relative log-real-wages of men with disabilities fell significantly by 5.2% at the 10% level. This result is in sharp contrast to wage estimates from using selfreported measure of disability (i.e., an insignificant increase by 3.3%).
Donohue et al. (2011):
we find little evidence of adverse effects on weeks worked but strong evidence of wage declines for the disabled, albeit declines beginning in 1986, well before the ADA’s passage. These results therefore cast doubt on the adverse ADA-related impacts found in previous studies, particularly Acemoglu and Angrist (2001). The conflicting narratives that emerge from our analysis shed new light on, but also counsel caution in reaching final conclusions about, the impact of the ADA on employment outcomes for people with disabilities.
The ADA appears to have been associated with a decrease in unexplained wage differentials both in absolute and relative terms, a part of which might be associated with economic discrimination. The empirical results do not show a similar trend for the unexplained part of the employment differential, which increased in absolute terms, and remained constant in relaitve terms in the post-ADA period. Overall, it seems discrimination on entry to employment is more important than wage discrimination for persons with disabilities and the ADA was less effective in reducing employment discrimination than wage discrimination.
Maroto & Pettinicchio (2014):
We found that the average employment rate for people with disabilities was higher in precisely those states with ADA-like laws, even after controlling for state spending. In addition, states that included provisions for reasonable accommodation also saw better employment outcomes. The same was true for earnings, although the relationship was weaker. These findings therefore mirror those of Beegle and Stock (2003). Given that people with disabilities fare better in progressive states with a longer history of disability antidiscrimination legislation, we suggest this is evidence against the unintended harms argument. Rather, our findings allude to the importance of antidiscrimination legislation in improving the economic well-being of people with disabilities. ...
Our court case findings were more ambiguous. We found that liberal Supreme Court rulings decreased employment levels for people with disabilities, and lower-level court settlements slightly increased earnings. Considering these relationships together, our findings indicate that, on the one hand, the creation of legislation has positively affected employment outcomes, but, on the other, the enforcement of legislation has had negative effects.
P.S. In response to the discussion in the comments above, Angrist is among the top economists and Acemoğlu is as sure a bet for the Economics Nobel in the next 10–20 years as any other economist. The Journal of Political Economy is one of the so-called top 5 journals in economics.
In contrast, the other authors cited here are far less well-known and their papers appeared in far less prestigious journals (in fact some were not published in any peer-reviewed journals).
The above information is just FYI; make what you will of it. I personally do not pay too much heed to how well-known an economist is or how prestigious a journal is.
P.P.S. There is too much faith in the peer review process in general and even here on this site. Those who actually have experience on how the sausages are made in the publication process will know that an article that's peer-reviewed isn't automatically "better" or less "error-prone" that one that isn't.