The generally accepted signal processing definition of a causal system is one in which its output does not respond to a given input before that input is applied. This means it has an impulse response that is zero for all time less than $t=0$. The phase can indeed be monotonically increasing for such a causal system. This will result in negative Group Delay over the portion of the frequency response where this occurs, but this does NOT mean the system is not causal. This interesting an unintuitive property is detailed in this other post.
The way to determine if a system is causal from its frequency response is to take the Hilbert Transform of the real part of the frequency response and compare it to the imaginary part. The negative of the Hilbert Transform of the real part of the frequency response must equal the imaginary part of the frequency response for a causal time domain impulse response.
For a causal system (one-sided in time):
$$H(\omega) = H_R(\omega) - j \hat{H}_R(\omega)$$
Where:
$H(\omega)$: Complex frequency response.
$H_R(\omega)$: Real portion only of complex frequency response.
$\hat{H}_R(\omega)$: Hilbert Transform of the real portion of the complex frequency response.