Update 2019
This is possible today. To monitor free disk space per drive with Azure Monitor do the following:
- Enable Guest Operating System OS Metrics for the VM.
 
- In the Azure Portal select the Virtual Machine.
 
- Click Diagnostics Settings (under Monitoring).
 
- Click the Performance counters tab.
 
- Click the Custom button.
 
- In the textbox add the custom metric for the drive you would like. e.g. 
\LogicalDisk(C:)\% Free Space. 
- Click Add and set the Unit to 
Percent. 
Source: Azure Support.
To view the logs from the Azure Guest Monitor for Linux:
// Virtual Machine free disk space
// Show the latest report of free disk space, per instance
InsightsMetrics
| where Name == "FreeSpacePercentage"
| summarize arg_max(TimeGenerated, *) by Tags
// arg_max over TimeGenerated returns the latest record
| project TimeGenerated, Computer, Val, Tags
This results in the following alert query (you need AggregatedValue and bin(TimeGenerated, <some time>) in the query):
InsightsMetrics
| where Name == "FreeSpacePercentage"
| summarize AggregatedValue=arg_min(Val, *)  by bin(TimeGenerated, 5min), Tags
To view the same for any generic diagnostics endpoint (thanks @gabe):
After turning this on, i was able to view the free disk space with a log query: 
// Virtual Machine free disk space 
// Show the latest report of free disk space, per instance 
Perf 
 | where ObjectName == "LogicalDisk" or 
// the object name used in Windows records 
  ObjectName == "Logical Disk" // the object name used in Linux records 
 | where CounterName == "Free Megabytes" 
 | summarize arg_max(TimeGenerated, *) by InstanceName 
// arg_max over TimeGenerated returns the latest record 
 | project TimeGenerated, InstanceName, CounterValue